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Random Numbers
Volume Number:8
Issue Number:3
Column Tag:Coding Efficiently

Fast Random Numbers

A random number generator that is 10 times faster

By Jon Bell, Clinton South Carolina

Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.

So-called “random numbers” are an important ingredient in many applications. This article discusses some of the theory behind random number generators, and shows how to replace the Toolbox random number generators with one that produces the same results but is more than ten times faster on a Mac SE/30.

[NOTE: The material in the following two sections is summarized from the article by Park and Miller which is listed in the references at the end of this article.]

Random Numbers

All random number generators produce sequences of numbers by using a fixed set of rules to produce each number in the sequence from its predecessors. This means that the numbers are not really “random” at all, in a technical sense, because they are completely predictable, provided that you know the rules. The only way to get true “random” numbers would be to use some physical process which is inherently random, such as rolling dice or counting the number of decays per second from a radioactive source. Nevertheless, a computer can produce numbers which “look” random to a person who doesn’t know the rule by which they were generated, and which satisfy many of the common statistical tests of randomness. Such numbers are often called pseudorandom numbers.

Depending on the application, there are many kinds of random number sequences. For example, we might want real (floating-point) numbers which lie between some given minimum and maximum value; or we might want integers, instead. We might want “uniformly” random numbers, in which the probability of getting any single number is the same as the probability of getting any other number; or we might want random numbers which are distributed according to some pattern, such as the familiar Gaussian (bell-shaped) curve. Any of these kinds of random numbers can be produced by applying a suitable transformation to a sequence of integers which is uniformly distributed between 1 and some maximum value, and so most random number generators have a random integer generator as their core.

Before actually looking at specific random number generators, it’s worth summarizing the desirable properties which they should have. First, we want to get as many different random numbers as possible, within the limits we specify. The problem is that any random number generator will begin to repeat itself eventually. If we calculate each number in the sequence based only on the previous number in the sequence, as soon as we generate a number we’ve generated before, the sequence will repeat itself from that point on. The length of the sequence before it begins to repeat is called its period. Clearly we would like the period to be as long as possible.

Second, the sequence of numbers, although completely deterministic, should “look” random enough for our purposes. At the very least, the numbers should be uniformly distributed, so that we are just as likely to get 1236 as we are to get 8490792, or 52804, or whatever. In addition, the numbers should not be correlated: if we take two (or three, or four...) numbers from the sequence according to some specified rule (say, two consecutive numbers), those numbers should not be “related” to each other in any “obvious” way. All random number generators can be made to fail this test, but some fail more “obviously” than others.

Finally, it should be possible to implement the generator correctly and (hopefully) efficiently on a wide variety of machines and in a wide variety of languages, without having to depend on “tricks” which are peculiar to a particular machine or language. No matter what the computing environment, we should be able to get repeatable results.

Many random number generators which have been implemented on various systems, or recommended in various programming textbooks, have failed to meet one or more of these criteria. Below I will describe a so-called “minimal standard” random number generator which meets all three to some extent. It is almost certainly not the “best” random number generator available, but it is “good enough” for most purposes, and can serve as a benchmark for evaluating other random number generators.

The “Minimal Standard” Generator

There are many different schemes for producing uniformly random integers on a computer. One of the simplest was devised by D.H. Lehmer in 1951. After choosing two parameters, the modulus (m) and the multiplier (a), and the first integer, z1, in the sequence, we generates successive numbers in the sequence as follows:

z2   =   az1 mod m
z3   =   az2 mod m
...
zn   =   azn-1 mod m

Here “mod” means “take the remainder after dividing by,” as in Pascal and other programming languages. By the nature of the mod operation, the result always lies between 0 and m-1, inclusive.

It turns out that if we let m = 231 - 1, there are 534,600,000 different multipliers, a, which give us a generator with the maximum possible period, namely 231 - 1. This is the longest period which we can attain using 32-bit integers.

It looks as if we have an embarassingly rich choice of multipliers. Unfortunately, most of them cannot be used correctly on a computer with 32-bit integers, because they would cause the intermediate product anz to overflow. In 1979, G. L. Schrage devised an implementation of the linear congruential generator which is less susceptible to integer overflow than the simple-minded version described above. Even with Schrage’s method, nevertheless, it turns out that only 23,093 of the 534,600,000 full-period multipliers can be used in a correct implementation. To decide which of these multipliers are “best”, we must study in detail the sequences they produce.

As far as randomness is concerned, linear congruential random number generators have a well-known flaw. If we take consecutive pairs of numbers, say (z1, z2), (z3, z4), (z5, z6), etc., from a linear congruential sequence, and plot them on a graph, we find that they invariably lie on a series of parallel diagonal lines. Similarly, consecutive triplets lie on a series of parallel planes in three-dimensional space; consecutive quadruplets lie on parallel hyperplanes in four-dimensional space; and so on. Depending on how we use the numbers, these correlations can occasionally produce strange non-random results. The spacing between the planes varies according to the choice of a and m, and should be made as small as possible.

In 1969, Lewis, Goodman and Miller designed a 32-bit random number generator for the IBM System/360. They decided to use a linear congruential generator with m = 231 - 1 = 2147483647 and a = 75 = 16807. Since then, this generator has been implemented on a variety of systems, and has been dubbed a “minimal standard” random number generator.

The “minimal standard” generator does not give the smallest hyperplane spacing, though. That honor appears to be shared by the two generators which use the multipliers a = 48271 and a = 69621. Nevertheless, I will use the “minimal standard” in my implementation. The program listings indicate which parameters you need to change if you want to experiment with the “better” multipliers.

Before looking at how to implement the minimal standard generator, though, let’s look at what the Macintosh Toolbox already provides us.

The QuickDraw Random Number Generator

The Macintosh Toolbox actually contains two random number generators. The first one is part of QuickDraw. Its seed is a QuickDraw global variable named randSeed, a longint which is initialized to 1 by InitGraf at the start of a program. The function Random updates the seed and returns an integer in the range from -32767 to 32767. I know nothing about the properties of this generator, but clearly its period must fall far short of the period of a decent 32-bit generator, and so I will not consider it further.

The SANE Random Number Generator

The Toolbox’s second random number generator is part of the SANE floating-point arithmetic package:

function RandomX (var x : extended) : extended;

The parameter x is a variable which keeps track of the sequence of random integers (which I called z in the discussion above). It is commonly called the seed for the random number generator. At the beginning of a program, we must initialize this variable to a suitable integer value in the range [1, 231 - 2], and make sure that we preserve its value between calls to RandomX. Whenever we call RandomX, it updates the seed to a new value, and, in addition, returns that value as its result. Note that the seed and result are always integer numbers, even though they are stored in extended-precision floating-point variables.

According to the Apple Numerics Manual, RandomX is in fact a “minimal standard” random number generator, as described above.

Having to preserve the value of the seed between calls to RandomX is a nuisance if you use RandomX in several different procedures whose scopes do not all overlap. One quick and easy solution is to make the seed a global variable, which, however, offends structured-programming purists like me. A better solution (in MPW or Think Pascal, anyway) is to set up a unit which declares the seed as a global variable in the implementation section. This effectively “hides” the seed from the rest of your program, and prevents you from trashing it accidentally in some other routine.

Listing 2 shows an MPW Pascal unit, SANERandomNumbers.p, which carries out this idea. The procedure InitRandomSeed initializes the seed to a specified value, and the parameterless function RandomSeed returns its value for inspection. These are the only two ways that another routine can gain access to the seed. The parameterless function RandomReal updates the seed and returns an extended-precision real number, uniformly distributed in the range (0,1). The function RandomInteger updates the seed and returns a longint, uniformly distributed in the range [1, max], where max is specified as a parameter.

Listing 1 shows a simple MPW tool which I used to test this generator, and the other ones to be described in this article. It initializes the seed to 1, displays the first ten random numbers in the sequence, displays the value of the seed after 10,000 iterations, and finally counts how many random real numbers are generated per second. (Actually, it counts for one minute, then divides by 60, to get a reliable average.) The 10,000th seed tests whether the generator has been implemented correctly; the minimal standard generator must produce the value 1043618065. If this value does not appear, either the generator is not the minimal standard one, or else it was not implemented correctly.

Likning this program with SANERandomNumbers gave the following results:

The first ten random numbers are:

0.000007826369259426
0.131537788143166242
0.755605322195033227
0.458650131923449287
0.532767237412169221
0.218959186328090348
0.047044616214486126
0.678864716868318951
0.679296405836612175
0.934692895940827623
After 10000 iterations,
the random seed is 1043618065.

In one second, 1381 random numbers were generated.

The “Minimal Standard” Generator in Pascal

Having to go through the Toolbox trap dispatcher slows down the SANE random number generator significantly, which is a drawback if we need to generate lots of random numbers quickly. Since we know the algorithm, why not simply implement it directly in Pascal and bypass the trap dispatcher completely? This would seem to be trivial: simply replace calls to RandomX with the statement

seed := (a * seed) mod m;

The problem is that the product a * seed can easily overflow the limits of a 32-bit integer variable, causing incorrect results from that point on. G. L. Schrage came to the rescue in 1979 with an algorithm which splits the seed into its high and low 16-bit halves. Schrage’s algorithm is implemented in the unit PasRandomNumbers.p, given in Listing 3. It is gratifyingly quicker than the SANE implementation, generating 3049 random reals per second versus 1381, a speedup of 2.2x. (These timings, and all those that follow, were measured on an unmodified Mac SE/30, using the program given in Listing 1.)

Schrage’s method has two hidden “bottlenecks” when implemented on a 68000 processor (Mac Plus and SE). The first one involves the operations seed div q and seed mod q, which require dividing a 32-bit integer by another 32-bit integer. On the 68000, the DIVS instruction can only divide a 32-bit quantity by a 16-bit quantity! Therefore a compiler must perform these operations in software. MPW Pascal inserts calls to the routines %I_DIV4 and %I_MOD4, which are located in the library Paslib.o. Similarly, the products a * lo and r * hi must be carried out as 32-bit multiplications, via the routine %I_MUL4 in Paslib.o.

If we tell the compiler to generate 68020/68030 code (in MPW Pascal, by using the -mc68020 option), these operations can each be performed with a single machine instruction. DIVS.L divides two 32-bit numbers to produce a 32-bit quotient, DIVSL.L divides two 32-bit numbers to produce both a 32-bit quotient and a 32-bit remainder, and MULS.L multiplies two 32-bit numbers to produce a 32-bit result. Recompiling PasRandomNumbers.p with the -mc68020 option allows us to produce 4491 random numbers per second, a speed increase of 1.5x over the preceding version.

In generating random real numbers (function RandomReal), there is another bottleneck in the final step, which is a floating-point division operation. If we do not instruct the compiler otherwise, it generates calls to the SANE package for all floating-point operations, which allows the code to run on all Macs, but is rather slow. If we are running on a Mac with a 68020 or 68030 CPU, we probably have a 68881 or 68882 floating-point coprocessor (FPU) available as well. (Exceptions include the Mac LC, and some accelerated SEs and Pluses.) Therefore, if we’re using the -mc68020 option, we may as well use -mc68881 as well, to tell the compiler to use the FPU for floating-point arithmetic. Recompiling PasRandomNumbers.p yet again, with both options enabled, allows us to produce 15067 random numbers per second, a speed of increase of 3.6x over the previous version. [Note: we must also recompile the main program with the -mc68881 option, because the FPU expects extended variables to be stored in 10 bytes, whereas SANE expects extended variables to be stored in 8 bytes.]

The “Minimal Standard” Generatorin

Assembly Language

Examining the compiler’s output for the version with 68020/30 and FPU code, we can (as usual) spot things which could be improved in hand-coded assembly language. In procedure UpdateSeed, there are two things in particular. First, the compiler uses separate operations for seed div q and seed mod q (DIVS.L and TDIVS.L respectively), even though TDIVS.L produces both the quotient and remainder, in separate registers. Second, there is quite a bit of “unnecessary” shuffling of data between registers.

The obvious solution here is to recode the entire unit in assembly language, which is not a very difficult task. The results are shown in Listing 4. To use it with the existing test program, we follow these steps:

• Assemble RandomNumbers.a to produce an object file RandomNumbers.a.o;

• Create an interface file, RandomNumbers.p, by removing the implementation section from PasRandomNumbers.p;

• Compile Test.p, using the interface file RandomNumbers.p to define the interfaces to the random-number routines. (Actually, I simply recycled Test.p.o from the previous test, since the interface to the random number-routines hadn’t changed);

• Link Test.p.o with RandomNumbers.a.o.

Results: 18758 random numbers per second, a speed increase of 1.2x over the previous version, or a net increase of 13.6x over the original SANE implementation!

For Fortran Freaks Only

Since many scientific applications are still written in Fortran, I ought to point out that any of the units described above can be easily used in Language Systems Fortran under MPW. Listing 5 shows a Fortran version of the test program given in Listing 1. LS Fortran compiles this to a standalone application rather than a MPW tool, but otherwise the two programs work the same way.

The most important thing to note here is that standard Fortran passes all subroutine arguments by reference, like Pascal’s var parameters. However, two of our random-number routines (InitRandomSeed and RandomInteger) expect their arguments to be passed by value. LS Fortran has an extension which enables this, in the form of the %VAL function:

CALL INITRANDOMSEED (%VAL(1))

This passes the value 1 directly to the subroutine, rather than creating a temporary variable, storing 1 in it, and passing the address of this temporary variable.

References

Apple Computer, Inc. Inside Macintosh, Volume I. Addison-Wesley, 1985.

Apple Computer, Inc. Apple Numerics Manual, 2nd ed. Addison-Wesley, 1988.

Stephen K. Park and Keith W. Miller. “Random Number Generators: Good Ones Are Hard to Find”, Communications of the ACM, vol. 31, p. 1192 (October 1988).

Steve Williams. 68030 Assembly Language Reference. Addison-Wesley, 1989.

Listing 1: Test.p
program TestRandomNumbers (input, output);

{  This program tests the random number generator imple-
   mented by the unit RandomNumbers.  If the generator is
   the "minimal standard generator" (multiplicative linear
   congruential algorithm with modulus 2147483647 and mul-
   tiplier 16807), and the initial seed is 1, the value of
   the seed after 10000 iterations should be 1043618065.  
   If this value is not obtained, then the generator either
   is not a "minimal standard" generator, or has failed due
   to integer overflow at some point.

   Under the Macintosh Programmer's Workshop, this program
   can be compiled and run as an MPW Tool.  Under other
   development systems, it may need some modification.

                           Jon Bell
                 Dept. of Physics & Comp. Sci.
                     Presbyterian College
                       Clinton SC 29325
                        CIS #70441,353                     }

USES RandomNumbers, Events;

var
k : longint;
x : extended;
stopTime : longint;

begin
{  First verify that the generator is working correctly.  }
InitRandomSeed (1);
writeln;
writeln ('The first ten random numbers are:');
writeln;
for k := 1 to 10 do
    writeln (RandomReal:20:18);
for k := 11 to 10000 do
    x := RandomReal;
writeln;
writeln ('After 10000 iterations,');
writeln ('the random seed is ', RandomSeed:1, '.');
writeln;
{  Now let's find out how fast the generator is. }
k := 0;
stopTime := TickCount + 3600;
while TickCount <= stopTime do
    begin
    x := RandomReal;
    k := k + 1
    end;
writeln ('In one second, ', trunc(k/60):1, 
         ' random numbers were generated.');
writeln
end.
Listing 2: SANERandomNumbers.p
UNIT RandomNumbers;

{  This unit provides a convenient interface to the SANE
   random number generator. }

INTERFACE

USES Sane;

procedure InitRandomSeed (newSeed : longint);

{  Initializes the random number seed to "newSeed".  You
   must call this procedure once, at the beginning of your
   program, before you use any of the following functions.
   As far as randomness is concerned, it makes no difference
   what value you use for "newSeed", so long as it isn't
   zero.  Using different seeds merely gives you different
   sequences of "random" numbers.  Using the same seed each
   time you run the program gives you the same sequence of
   "random" numbers each time, which may be useful for
   debugging. }

function RandomSeed : longint;

{   Returns the current value of the random number seed.   }

function RandomReal : extended;

{  Returns a real number, "randomly" selected from the
   interval [0,1].                                         }

function RandomInteger (max : longint) : longint;

{  Returns an integer, "randomly" selected from the
   interval [1,max].                                       }

IMPLEMENTATION

const
m = 2147483647;

var
seed : extended;

procedure InitRandomSeed (newSeed : longint);

begin
seed := newSeed
end;

function RandomSeed : longint;

begin
RandomSeed := round(seed)
end;

function RandomReal : extended;

begin
RandomReal := RandomX (seed) / m;
end;

function RandomInteger (max : longint) : longint;

var
ignore : extended;

begin
ignore := RandomX (seed);
RandomInteger := (round(seed) mod max) + 1
end;

END.
Listing 3: PasRandomNumbers.p
UNIT RandomNumbers;

{  This unit implements a "minimal standard" random number 
   generator. }

INTERFACE

procedure InitRandomSeed (newSeed : longint);

function RandomSeed : longint;

function RandomReal : extended;

function RandomInteger (max : longint) : longint;

IMPLEMENTATION

const
a = 16807;
m = 2147483647;
q = 127773;        { m div a }
r = 2836;          { m mod a }

{  NOTE:  Other possible values for these constants, which
   may give "better" results, are:

      a = 48271,  q = 44488,  r = 3399  or
      a = 69621,  q = 30845,  r = 23902,

   keeping m = 2147483647.                                 }

var
seed : longint;

procedure InitRandomSeed (newSeed : longint);

begin
seed := newSeed
end;

function RandomSeed : longint;

begin
RandomSeed := seed
end;

{private} procedure UpdateSeed;

var
lo, hi, test : longint;

begin
hi := seed div q;
lo := seed mod q;
test := a * lo - r * hi;
if test > 0 then
    seed := test
else
    seed := test + m
end;

function RandomReal : extended;

begin
UpdateSeed;
RandomReal := seed / m
end;

function RandomInteger (max : longint) : longint;

begin
UpdateSeed;
RandomInteger := (seed mod max) + 1
end;

END.
Listing 4: RandomNumbers.a
 MACHINE MC68020
         MC68881

;-----------------------------------------------------------
;  WARNING:  These routines require a 68020 or 68030 CPU,
;  and a 68881 or 68882 floating-point coprocessor!!!
;-----------------------------------------------------------
;  Numerical constants used in the seed-updating algorithm.

a           EQU         16807       ; multiplier
m           EQU         2147483647  ; modulus
q           EQU         127773      ; m div a
r           EQU         2836        ; m mod a

;-----------------------------------------------------------
;  The random number seed (a 32-bit integer) is a global
;  variable whose value is preserved between calls to
;  the random number functions.

Seed        DS.L  1

InitRandomSeed    PROC        EXPORT

;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
;  Initializes the random number seed to a specified value.
;
;  Pascal interface:
;      procedure InitRandomSeed (newSeed : longint);
;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

newSeed     EQU         8
ParamSize   EQU         4
LocalSize   EQU         0

            LINK        A6, #LocalSize
            MOVE.L      newSeed(A6), seed(A5)
            UNLK        A6
            RTD         #ParamSize

            DC.B        'INITRANDOMSEED'

RandomSeed        FUNC        EXPORT

;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
;  Returns the current value of the random number seed.
;
;  Pascal interface:
;      function RandomSeed : longint;
;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

result      EQU         8
LocalSize   EQU         0

            LINK        A6, #LocalSize
            MOVE.L      Seed(A5), result(A6)
            UNLK        A6
            RTS

            DC.B        'RANDOMSEED'

UpdateSeed        PROC

;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
;  Applies the Lehmer / Lewis-Goodman-Miller / Schrage
;  algorithm to update the random number seed.  This
;  procedure is not to be used outside this unit, therefore
;  it is not EXPORTed.
;
;  It stores the new seed in the global variable "Seed",
;  and leaves a copy in register D1 for use by the calling
;  routine.
;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

hi          EQU         D0
lo          EQU         D1

            MOVE.L      Seed(A5), hi
            TDIVS.L     #q, lo:hi
            MULS.L      #a, lo
            MULS.L      #r, hi
            SUB.L       hi, lo            
            BGT.S       StoreNewSeed
            ADD.L       #m, lo            ; if lo <= 0

StoreNewSeed

            MOVE.L      lo, Seed(A5)
            RTS

            DC.B        'UPDATESEED'

RandomReal        FUNC        EXPORT

;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
;  Updates the random number seed and returns a real number
;  in the range (0,1).
;
;  Pascal interface:
;      function RandomReal : extended;
;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

quotient    EQU         FP0
newSeed     EQU         D1
result      EQU         8
LocalSize   EQU         0

            LINK        A6, #LocalSize
            JSR         UpdateSeed
            FMOVE.L     newSeed, quotient
            FDIV.L      #m, quotient
            FMOVE.X     quotient, ([result,A6])
            UNLK        A6
            RTS

            DC.B        'RANDOMREAL'

RandomInteger     FUNC        EXPORT

;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
;  Updates the random number seed and returns a longint
;  in the range [1,max].
;
;  Pascal interface:
;      function RandomInteger (max : longint) : longint;
;- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

dividend    EQU         D1
divisor     EQU         D0
result      EQU         8
max         EQU         12
LocalSize   EQU         0
ParamSize   EQU         4

            LINK        A6, #LocalSize
            JSR         UpdateSeed

            ; The new seed is now in "dividend."

            TDIVS.L     max(A6), divisor:dividend

            ; "Divisor" now contains the remainder.

            ADDQ.L      #1, divisor
            MOVE.L      divisor, result(A6)
            UNLK        A6
            RTD         #ParamSize

            DC.B        'RANDOMINTEGER'

            END
Listing 5: TestF77.f
C-----------------------------------------------------------
C  This program demonstrates the use of a hand-coded 
C  random-number generator in a Language Systems Fortran
C  program.  It can be linked with any of the following
C  object files:
C
C  SANERandomNumbers.p.o (SANE's random number generator)
C  PasRandomNumbers.p.o (hand-coded in Pascal)
C  RandomNumbers.a.o (hand-coded in assembly language)
C-----------------------------------------------------------
!!M Inlines.f
      IMPLICIT NONE
      INTEGER K, STOPTIME, RANDOMSEED
      EXTENDED X, RANDOMREAL
      EXTERNAL RANDOMREAL, RANDOMSEED
      CALL INITRANDOMSEED (%VAL(1))
      PRINT *
      PRINT *, 'The first ten random numbers are:'
      PRINT *
      DO K = 1, 10
         PRINT '(1X, F20.18)', RANDOMREAL()
      END DO
      DO K = 11, 10000
         X = RANDOMREAL()
      END DO
      PRINT *
      PRINT *, 'After 10000 iterations,'
      PRINT *, 'the random seed is ', RANDOMSEED(), '.'
      PRINT *
      K = 0
      STOPTIME = TICKCOUNT() + 3600
      DO WHILE (TICKCOUNT .LE. STOPTIME)
         X = RANDOMREAL()
         K = K + 1
      END DO
      PRINT *, 'In one second, ', K/60, 
     &         ' random numbers were generated.'
      PRINT *
      END

 

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There’s a lot going on right now, and I don’t really feel like trying to write some kind of pithy intro for it. All I’ll say is lots of people have been coming together and helping each other in small ways, and I’m choosing to focus on that as I... | Read more »
Genshin Impact Guide - Gacha Strategy: W...
If you're playing Genshin Impact without spending money, you'll always need to be looking for ways to optimize your play to maximize rewards without getting stuck in a position where you're tempted to spend. The most obvious trap here is the game'... | Read more »
Genshin Impact Adventurer's Guide
Hello and well met, fellow adventurers of Teyvat! Check out our all-in-one resource for all things Genshin Impact. We'll be sure to add more as we keep playing the game, so be sure to come back here to check for updates! [Read more] | Read more »
Genshin Impact Currency Guide - What...
Genshin Impact is great fun, but make no mistake: this is a gacha game. It is designed specifically to suck away time and money from you, and one of the ways the game does this is by offering a drip-feed of currencies you will feel compelled to... | Read more »
XCOM 2 Collection on iOS now available f...
The XCOM 2 Collection, which was recently announced to be coming to iOS in November, is now available to pre-order on the App Store. [Read more] | Read more »
Presidents Run has returned for the 2020...
IKIN's popular endless runner Presidents Run has returned to iOS and Android just in time for the 2020 election season. It will see players choosing their favourite candidate and guiding them on a literal run for presidency to gather as many votes... | Read more »
New update for Cookies Must Die adds new...
A new update for Rebel Twins’ platformer shooter Cookies Must Die is coming out this week. The update adds quite a bit to the game, including new levels and characters to play around with. [Read more] | Read more »
Genshin Impact Guide - How to Beat Pyro...
The end game of Genshin Impact largely revolves around spending resin to take on world bosses and clear domain challenges. These fights grant amazing rewards like rare artifacts and ascension materials for weapons and adventurers, but obviously... | Read more »

Price Scanner via MacPrices.net

Use our exclusive iPhone Price Trackers to fi...
Looking for a new Apple iPhone 12 or 12 Pro? Perhaps a deal on last year’s iPhone 11? Check out our iPhone Price Tracker here at MacPrices.net. We track new and clearance iPhone prices from Apple as... Read more
Weekend deal: $100 off 13″ MacBook Airs at Am...
Amazon has new 2020 13″ MacBook Airs on sale for $100 off Apple’s MSRP, starting at only $899. Their prices are the lowest available for new MacBooks from any Apple resellers. These are the same 13″... Read more
New 10.9″ 64GB Apple iPad Air on sale for $55...
Amazon has Apple’s new 2020 10.9″ 64GB WiFi iPad Air on sale today for $549.99 shipped. That’s $40 off MSRP. Pre-orders are available today at this discounted price, and Amazon states that the iPad... Read more
Get a clearance 2019 27″ 5K iMac for up to $5...
Apple has Certified Refurbished 2019 27″ 5K iMacs available starting at $1439 and up to $520 off their original MSRP. Apple’s one-year warranty is standard and shipping is free. The following... Read more
AT&T offers the Apple iPhone 11 for $10/m...
AT&T is offering Apple’s 64GB iPhone 11 for $10 per month, for customers opening a new line of service, no trade-in required. Discount is applied via monthly bill credits over a 30 month period.... Read more
Apple’s 2020 11″ iPad Pros on sale today for...
Apple reseller Expercom has new 2020 11″ Apple iPad Pros on sale for $50-$75 off MSRP, with prices starting at $749. These are the same iPad Pros sold by Apple in their retail and online stores: – 11... Read more
Did Apple Drop The Ball By Not Branding Its C...
EDITORIAL: 10.21.20 – In the branding game, your marketing strategy can either be a hit or a miss and the latter is the case for Apple when it missed out on an opportunity to brand its “SE” series of... Read more
27″ 6-core and 8-core iMacs on sale for up to...
Adorama has Apple’s 2020 27″ 6-core and 8-core iMacs on sale today for $50-$100 off MSRP, with prices starting at $1749. Shipping is free: – 27″ 3.1GHz 6-core iMac: $1749, save $50 – 27″ 3.3GHz 6-... Read more
Apple’s 16″ MacBook Pros are on sale for $300...
B&H Photo has 16″ MacBook Pros on sale today for $300-$350 off Apple’s MSRP, starting at $2099. Expedited shipping is free to many addresses in the US. Their prices are among the lowest available... Read more
Apple has 2020 13″ MacBook Airs available sta...
Apple has a full line of Certified Refurbished 2020 13″ MacBook Airs available starting at only $849 and up to $200 off the cost of new Airs. Each MacBook features a new outer case, comes with a... Read more

Jobs Board

Dental Receptionist - *Apple* Valley Clinic...
Dental Receptionist - Apple Valley Clinic + Job ID: 57314 + Department: Apple Valley Dental + City: Apple Valley, MN + Location: HP - Apple Valley Clinic Read more
*Apple* Mobility Specialist - Best Buy (Unit...
**788165BR** **Job Title:** Apple Mobility Specialist **Job Category:** Store Associates **Store Number or Department:** 001013-Virginia Commons-Store **Job Read more
Cub Foods - *Apple* Valley - Now Hiring Par...
Cub Foods - Apple Valley - Now Hiring Part Time! United States of America, Minnesota, Apple Valley Retail Post Date Oct 08, 2020 Requisition # 124800 Sign Up for Read more
*Apple* Mobility Specialist - Best Buy (Unit...
**784631BR** **Job Title:** Apple Mobility Specialist **Job Category:** Store Associates **Store Number or Department:** 000522-Baxter-Store **Job Description:** The Read more
Senior Data Engineer - *Apple* - Theorem, L...
Job Summary Apple is seeking an experienced, detail-minded data engineeringconsultant to join our worldwide business development and strategy team. If you are Read more
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