Better function approximations: Taylor vs. Remez

You may have once crossed this particular piece of magic:

$\sin(a) = a - \dfrac{a^3}{3!} + \dfrac{a^5}{5!} - \dfrac{a^7}{7!} + \dfrac{a^9}{9!} + \dots$

The right part is the Taylor series of sin around 0. It converges very quickly to the actual value of sin(a). This allows a computer to compute the sine of a number with arbitrary precision.

And when I say it’s magic, it’s because it is! Some functions, called the entire functions, can be computed everywhere using one single formula! Other functions may require a different formula for different intervals; they are the analytic functions, a superset of the entire functions. In general, Taylor series are an extremely powerful tool to compute the value of a given function with very high accuracy, because for several common functions such as sin, tan or exp the terms of the series are easy to compute and, when implemented on a computer, can even be stored in a table at compile time.

Approximating sin with Taylor series

This is how one would approximate sin using 7 terms of its Taylor series on the [-π/2, π/2] interval. The more terms, the better the precision, but we’ll stop at 7 for now:

static double taylorsin(double x)
    static const 
    double a0 =  1.0,
           a1 = -1.666666666666666666666666666666e-1,  /* -1/3! */
           a2 =  8.333333333333333333333333333333e-3,  /*  1/5! */
           a3 = -1.984126984126984126984126984126e-4,  /* -1/7! */
           a4 =  2.755731922398589065255731922398e-6,  /*  1/9! */
           a5 = -2.505210838544171877505210838544e-8,  /* -1/11! */
           a6 =  1.605904383682161459939237717015e-10; /*  1/13! */
    double x2 = x * x;
    return x * (a0 + x2 * (a1 + x2 * (a2 + x2
             * (a3 + x2 * (a4 + x2 * (a5 + x2 * a6))))));

And you may think…

/raw-attachment/blog/2011/12/14/understanding-motion-in-games/derp.png “Oh wow that is awesome! So simple for such a difficult function. Also, since I read your masterpiece about polynomial evaluation I know how to improve that function so that it is very fast!”

Well, actually, no.


If you are approximating a function over an interval using its Taylor series then either you or the person who taught you is a fucking idiot because a Taylor series approximates a function near a fucking point, not over a fucking interval, and if you don’t understand why it’s important then please read on because that shit is gonna blow your mind.

Error measurement

Let’s have a look at how much error our approximation introduces. The formula for the absolute error is simple:

$\text{E}(x) = \left|\sin(x) - \text{taylorsin}(x)\right|$

And this is how it looks like over our interval:

You can see that the error skyrockets near the edges of the [-π/2, π/2] interval.

/raw-attachment/blog/2011/12/14/understanding-motion-in-games/derp.png “Well the usual way to fix this is to split the interval in two or more parts, and use a different Taylor series for each interval.”

Oh, really? Well, let’s see the error on [-π/4, π/4] instead:

I see no difference! The error is indeed smaller, but again, it becomes extremely large at the edges of the interval. And before you start suggesting reducing the interval even more, here is the error on [-π/8, π/8] now:

I hope this makes it clear that:

  • the further from the centre of the interval, the larger the error
  • the error distribution is very unbalanced
  • the maximum error on [-π/2, π/2] is about 6.63e-10

And now I am going to show you why that maximum error value is pathetic.

A better approximation

Consider this new function:

static double minimaxsin(double x)
    static const
    double a0 =  1.0,
           a1 = -1.666666666640169148537065260055e-1,
           a2 =  8.333333316490113523036717102793e-3,
           a3 = -1.984126600659171392655484413285e-4,
           a4 =  2.755690114917374804474016589137e-6,
           a5 = -2.502845227292692953118686710787e-8,
           a6 =  1.538730635926417598443354215485e-10;
    double x2 = x * x;
    return x * (a0 + x2 * (a1 + x2 * (a2 + x2
             * (a3 + x2 * (a4 + x2 * (a5 + x2 * a6))))));

It doesn’t look very different, right? Right. The values a0 to a6 are slightly different, but the rest of the code is strictly the same.

Yet what a difference it makes! Look at this error curve:

That new function makes it obvious that:

  • the error distribution is better spread over the interval
  • the maximum error on [-π/2, π/2] is about 4.96e-14

Check that last figure again. The new maximum error isn’t 10% better, or maybe twice as good. It is more than ten thousand times smaller!!

The minimax polynomial

The above coefficients describe a minimax polynomial: that is, the polynomial that minimises a given error when approximating a given function. I will not go into the mathematical details, but just remember this: if the function is sufficiently well-suited (as sin, tan, exp etc. are), then the minimax polynomial can be found.

The problem? It’s hard to find. The most popular algorithm to find it is the Remez exchange algorithm, and few people really seem to understand how it works (or there would be a lot less Taylor series). I am not going to explain it right now. Usually you need professional math tools such as Maple or Mathematica if you want to compute a minimax polynomial. The Boost library is a notable exception, though.

But you saw the results, so stop using Taylor series. Spending some time finding the minimax polynomial is definitely worth it. This is why I am working on a Remez framework that I will make public and free for everyone to use, modify and do what the fuck they want. In the meantime, if you have functions to numerically approximate, or Taylor-based implementations that you would like to improve, let me know in the comments! This will be great use cases for me.

Attachments (4)

Download all attachments as: .zip


1. anonymous -- 2011-12-26 21:19

Hi, Sam,

I searched high and low for high precision (not actually arbitrary, 32 digits would be enough for me now) approximation coeffs to the usual bunch of functions (trigonometric, hyperbolic and their inverses) and I found your blog after I have read the rest of the whole Net... :-) I don't have Mathematica or Maple (SciLab yes but it doesn't seem to have support for this application of Remez). I could compile Boost if you tell me there is no other alternative but do you tell me? :-) I don't think your framework is mature enough to use, isn't it?



2. anonymous -- 2011-12-27 20:20

Hi, Sam,

I visited a friend with a Matlab and, dear me, it can't do more than binary double, amounting to some 16 digits. Sheesh...



3. sam -- 2011-12-29 19:53

@Gábor: Well you will be happy to learn that I made a first release of LolRemez. I hope it can prove helpful!

4. anonymous -- 2011-12-31 22:04

Definitely, thanks, very interested, I'll check it out first thing in the next year... :-D

I'm also interested in the real package you use there. Right now I use 128-bit decimal floats (based on the DecNumber code) and I'm quite satisfied with it (actually, this is not gaming, speed is not of the utmost importance here, but it seems to be rather optimized maybe even for that). Functions are mostly Taylor now, because I needed to address other issues more urgently but the tangent family, as obvious, is very poor. I already started to calculate some simpler Chebyshev expansions but now I'll make an effort to use your Remez solution.

5. anonymous -- 2012-01-01 18:27

With the order above 20, I started to get real printouts like 10.000000000000000000000-1-3-4-2-8-5-6-2-6-6-2-1-9-6-6-50-9e-1. Now I modify my real package to work with your Remez implementation. And I don't really like the >> operator, by the way, I don't think it's elegant to have that with a real... :-D

Bye, Gábor

6. sam -- 2012-01-02 04:16

@Gábor: thanks for the feedback. The printout issues were a bug that is fixed in changelist [1121]. I agree about the >> operator; I removed it in changelist [1122] and made the integer division and multiplication detect when the operand is a power of two.

These changes will be in LolRemez 0.2 but you can probably apply them manually if you need them and don’t want to wait.

7. Gábor -- 2012-01-03 12:12

Yes, it started to behave very nicely now. If only the same could be said of inverse trigonometric functions... Those pesky near-one values ruin everything, as usual... :-)

8. sam -- 2012-01-03 14:36

@Gábor: usually inverse trigonometric functions use the following identity:

\[\arcsin(x) = \dfrac{1}{2}-2 \arcsin\sqrt{\dfrac{1-x}{2}}\]

The downside is that you need to compute a square root. The upside is that the arcsin polynomial coefficients don't change, so there is little memory access overhead.

9. Gábor -- 2012-01-04 12:27

@Sam, I only want to get arctan now, the other two can be calculated then using the half angle formula. And I also want to experiment with the Euler arctan approximation formula now. With factorials and powers it seems to offer better convergence. I didn't have time to work on it yesterday but I'll check it out now.

Actually, even with simple sin and cos, in this precision range, Remez is not that much better than the naive Taylor. I could reach my machine epsilon with a 25th order Remez while naive Taylor requires some 30-35 iterations. Still some gain, of course, but not *that* spectacular. :-) But I do stress the precision range, the relative gain is nicer if you only use, say, double binary FP. But sin and cos behave themselves, of course, not like their inverse functions.

I have all other functions in the decNumber package, both logs, exp, pow, sqrt, so there is no problem there.

10. Gábor -- 2012-01-09 14:18

@Sam, actually, I solved everything based on Euler for atan, thanks, it works just fine for my purposes now.

11. sam -- 2012-01-10 01:06

@Gábor: I am sorry but I realised that an accuracy of 1e-18 was hardcoded in LolRemez 0.1. Usually this isn’t a problem because people target float or double. But if your machine epsilon is smaller, I recommend you use LolRemez 0.2 instead. Also, convergence is now significantly faster. Sorry for the inconvenience!

By the way, can I quote you on “I could reach my machine epsilon with a 25th order Remez while naive Taylor requires some 30-35 iterations. Still some gain, of course, but not *that* spectacular.”? I find this quite funny :-)

12. sam -- 2012-01-11 01:18

@Sam. Don't. :-)) I wouldn't be that liberal with it if I was writing games or any other piece of software that needs to call it repeatedly, of course. But in this very particular case, it is calculated once per user input and, frankly, 5 ms versus 10 ms isn't a difference you'd really notice in this case. So, I only wanted to stress that unlike with sin/cos, doing something to avoid Taylor with atan was really indispensable because even 10,000 iterations (for a macheps of 1e-34) couldn't make it. That's still a bit larger for a margin, isn't it? :-)

But, actually, 1e-34 is crazy, of course. No real need for it, either, just pure marketing... :-)

13. andrew -- 2012-02-08 22:52

Hey Sam, you're a lifesaver! I've been searching the net for function approximation software for two days now - a lot a paper and a lot of talk - but not a lot of people seem to actually have the guts to do it! Thanks! I'm actually just using your code just to generate coefficients for waveshaping functions for real-time audio (atm) where approximation is a godsend since 4x Oversampling is used in general. Even with the odd bug here and there - it's still functional - and this is coming from a math and programming noob who has only started learning C++ two weeks ago. And yeah - The Taylor Series sucks balls!

Regards Andrew Ainslie South Africa

14. anonymous -- 2012-02-09 13:52

Found your website through the kvraudio developer forum. Heya, I tried lolremez0.2 and tanh(x) with 8 coefficients error 1.5266e-4 then I tried tanh's identity sinh(x)/cosh(x)- error 8.1442e-5...

Random programming thing - how do I add numbers to the formula? i.e. sin(2*x)? Getting compile errors here if I add numbers - probably missing something obvious...

15. sam -- 2012-02-09 18:07

@Andrew: I think not everything is implemented for mixed real/float operations, but if 2*x doesn't work, x*2 will probably.

Until I have implemented every operator, you can just cast the number to real: sin(real(2) * x). Also check out the bottom of real.h, it has predefined values for π, e, etc.

I'll check your tanh result to see what may cause the difference.

16. andrew -- 2012-02-09 21:52

Hey Sam! Thanks for the help: x*2 does work.

Haven't tried casting the number yet. Also did a little experiment: tried the Gudermannian function two ways:

R_PI_2*(atan(tanh((x*R_2_SQRTPI)/2))) - this didn't work

then I tried:

(atan(tanh((x*1.1283791)/2)))*1.5707963 - this did work

Inanyway - thanks for the great software! It has already saved me a great deal of cycles. Looking forward to future development.

Regards Andrew

17. sam -- 2012-02-13 22:22

@Andrew: real is a class, not a namespace. In order to use R_PI_2 and the like, you need to write real::R_PI_2.

18. ChriSopht -- 2012-02-18 18:23

Thanks for your great software! I wanted to replace some Taylor-polynomials in my code for a long time, but always was too lazy to implement a Remez algorithm myself ... Your code really works like a charm, very easy to approximate own functions.

Anyways I have some small suggestions:

Make the output routine independent from the calculation routine and maybe make it possible to get the coefficients manually. I had to adapt your code, to print output as "{ a0, a1, ...}" to directly copy'n'paste it to C++.

Furthermore, your program might be slightly faster, if you don't calculate the inverse of your matrix explicitly, but do matrix-decomposition and solving (not really critical, as no-one cares if it runs in 10 or 15 secs ...)

Also what might be nice is support for rational approximations as well. If one needs to approximate atan(y/x) there is a division anyways, so rational polynomials might be better.

19. sam -- 2012-02-21 18:56

@ChriSopht: thanks for the suggestions. I'll make it easier to gather and format the coefficients in the next release.

Also, I think it's easy to understand the reason why I did not bother to improve the matrix inversion: the running time of one inversion is less than two milliseconds! The search for local extrema is clearly what needs to be improved first, and I will work on that.

I really would like to support rational approximations, too. The biggest problem is the increased instability: the algorithm is not guaranteed to converge at all.

20. ChriSopht -- 2012-02-23 01:15

@sam: W.r.t runtime: ok, I never bother to benchmark your program, maybe it will only start being an issue if the polynomial gets very big.

W.r.t rational approximations, maybe the best way would be to make it some way interactive -- similar to the way boost's minimax optimizer does it. The basic idea they apply is to let the user manually skew the initial critical points towards or away from zero -- of course this only makes sense in an interactive version. Actually, I also like that they make it possible to decide for number of coefficients and precision of calculation at runtime. A big disadvantage of boost is at the moment that the weight is limited to relative or absolute and only the first coefficient can (directly) be pinned to zero (or to any other constant by shifting the function). Of course tricks such as y=sqrt(x) and sin(y)/y can be used as well, however "relative weight" then means relative to that modified function, not the original.

21. Tomas -- 2012-05-25 07:35

Hi Sam,

I will take the bait on "If you are approximating a function over an interval using its Taylor series..." part. As far as I remember, I was taught to do that, with a single additional step of approximating the error term as a power of x or sum of few of those, and approximating it using Chebyschev polynomial. I would say this is the Taylorian baseline to compare against. (I was not doing it for some time, but I may try that over weekend for 7 terms of sinus just to have the feeling)

22. mz -- 2012-12-22 13:04

Hi, interesting thread. For the specific case of the arctangent the following two rational approximations might be of your interest:

[This is for 0 <= x < Infinity. When x < 0 just take -atan_approx(-x)]

Second order:

atan_approx(x) = (Pi/2)*(b*x + x*x)/(1 + 2*b*x + x*x) where b = 0.596227, with a maximum approximation error of 0.1620º

Third order:

atan_approx(x) = (Pi/2)*(c*x + x*x + x*x*x)/(1 + (c+1)*x + (c+1)x*x + x*x*x) where c = (1 + sqrt(17))/8 and the maximum approximation error is 0.00811º

23. sam -- 2012-12-22 13:17

@mz Interesting, thanks. I would love for LolRemez to handle rational approximations, too, but the Remez algorithm is no longer guaranteed to converge in that case. I'll need to first find the relevant scientific papers — and there aren't many, unfortunately.

24. -- 2015-02-22 20:41

Nice post. I have been playing with your impl to approximate the normalized sinc function. I am breaking the intervals at the zeros, seems to be the best approach. Thanks for the work, it's been educational/helpful!

25. anonymous -- 2015-05-31 09:53

aren't the numbers 6e-10, 2e-14, 1e-19 very small? what am I missing here?

26. -- 2016-06-15 19:32

Wrong comparison!!!

Your first example is: with 6 coefficients a1 to a6, the Taylor series gets you worse error at the ends, worse by a factor of 10,000. In other words, you're choosing the number of coefficients and comparing the worst-case errors.


What you SHOULD be comparing is: given a target worst-case error (e.g. 6.0E-10), how many coefficients are needed for each of the two methods (Taylor v. minimax)?

I assume the number of coefficients gives a pretty strong indication of the execution time. In other words, you should not be giving the computer a time limit and asking what it can do; you should be giving it a worst tolerable error and asking how long it will take.

It's also worth noting that, in specifying the interval of interest, you'd better not try to use the resulting minimax polynomial outside that interval, because the polynomial shoots off like crazy. The higher the order, the more quickly it shoots off. It's worth making that explicit because it's not obvious from your graph of minimax error.

In this case of sine and cosine, I have found substantially reduced computation time with the following recipe:

  1. Divide the angle by a power of 2
  1. calculate sine and cosine using a short Taylor series (small angle, ergo smaller error); the relative error here determines the relative error of the final result: use this to choose how many terms to compute
  1. use the double-angle formulae the right number of times:

sin 2x = 2 sinx.cosx and cos 2x = 1 - 2(sinx)2

The above formula for cos2x is better from an error-propagation point of view than cos 2x = -1 + 2.(cosx)2

By optimising the size of the initial power series v. the number of halvings/doublings, you get more accuracy with less computation. That's because you have two processes here (doubling/halving and power series) letting you choose how much work to perform in each, instead of only one process (power series). That gives you freedom to find the right balance which gets the smallest error for any quantity of computational work.

For small required accuracies (10 decimal digits) there's not much in it, but when you get to larger numbers of digits, this method gets far more powerful. The general trend is that the amount of computation required (i.e. number of multiplies) is proportional to the square root of the number of digits required.

None of the above prevents you reducing the angle to the 0 - 45 degrees range (or the equivalent in radians) and of course you'd need an accurate value of pi for that.

27. Marc Reynolds -- 2016-08-10 09:44

Power series are always at a single point of evaluation, the error bound always grows away from that point. The various minimax methods are for minimizing an error measure across an interval.

39. MaoisesDAK -- 2017-07-12 21:51

personal loans for bad credit in lexington ky - loans for bad credit

<a href=""> pay day loans</a>

cash advance in athens tn

42. ElleDAK -- 2017-07-12 22:51

payday loan ceres - payday loans online

<a href=""> personal loans</a>

pay day loans capped

43. MaoisesDAK -- 2017-07-12 22:51

quick cash loans company - payday express

<a href="">payday loans</a>

short term loans no credit history

47. ElleDAK -- 2017-07-12 23:52

cash advance loan store - payday loans online

<a href=""> cash advance</a>

small loans brewton al

52. ElleDAK -- 2017-07-13 00:47

3000 dollar loan for bad credit - cash advance

<a href=""> personal loans</a>

payday loans in horizon

54. johhnycem -- 2017-07-13 01:31

ex6s4tpeta3y7roix5 <a href="">google</a> <a href=>google</a> 9itp872rue5cvtnu75

59. MaoisesDAK -- 2017-07-13 04:38

get money in 24 hours - payday loans online

<a href="">fast payday loans</a>

bad credit personal loans for 5000 edmonton

63. ElleDAK -- 2017-07-13 05:59

payday loans sector - payday loans online

<a href=""> payday express</a>

best cash advance Newport News Virginia

72. MaoisesDAK -- 2017-07-13 10:00

no teletrack cash advance lenders - cash advance

<a href=""> payday loans online</a>

payday loan eden nc

144. CharlesTew -- 2017-07-17 04:44

wh0cd659461 <a href=>Generic Triamterene</a> <a href=>ORDER CLONIDINE</a>

149. AlfredKer -- 2017-07-17 12:51

wh0cd732930 <a href=>lipitor</a> <a href=>buy prednisone online without a script</a> <a href=>proscar</a> <a href=>prednisolone</a> <a href=>medrol</a> <a href=>tadalafil 20 mg</a>

150. CharlesTew -- 2017-07-18 06:50

wh0cd924734 <a href=>clomid</a> <a href=>abilify generic</a> <a href=>acyclovir</a> <a href=>tamoxifen</a>

151. CharlesTew -- 2017-07-18 08:31

wh0cd777781 <a href=>mentax</a> <a href=>dostinex</a> <a href=>buy nizoral</a> <a href=>lasuna</a> <a href=>amantadine</a> <a href=>example</a> <a href=>buy reglan online</a>

152. CharlesTew -- 2017-07-18 12:51

wh0cd998204 <a href=>detrol</a> <a href=>get more information</a> <a href=>generic duricef</a> <a href=>actonel price</a>

153. CharlesTew -- 2017-07-19 01:44

wh0cd528745 <a href=>Buy Fluoxetine</a> <a href=>acyclovir w/o prescription</a> <a href=>furosemide 40mg</a>

154. CharlesTew -- 2017-07-19 08:33

wh0cd969596 <a href=>diclofenac</a> <a href=>nexium</a> <a href=>NEXIUM SALE</a>

155. AlfredKer -- 2017-07-19 15:21

wh0cd500130 <a href=>Valtrex</a>

156. CharlesTew -- 2017-07-19 15:31

wh0cd279695 <a href=>generic retin a</a>

157. CharlesTew -- 2017-07-20 06:52

wh0cd940971 <a href=>DICLOFENAC 50 MG</a>

158. KennethCeali -- 2017-07-20 10:01

wh0cd691920 <a href=>prilosec from india</a> <a href=>buy provera pills</a> <a href=>generic celexa</a>

159. AlfredKer -- 2017-07-20 17:16

wh0cd618457 <a href=>buy amitriptyline</a> <a href=>tadalafil</a> <a href=>phenergan cost</a> <a href=>generic vardenafil</a>

160. AlfredKer -- 2017-07-20 20:03

wh0cd324557 <a href=>evista</a> <a href=>medrol</a> <a href=>cheap zithromax</a> <a href=>nolvadex tamoxifen</a> <a href=>albuterol</a> <a href=>amoxicillin</a> <a href=>tricor</a> <a href=>inderal</a>

161. CharlesTew -- 2017-07-20 21:51

wh0cd544976 <a href=>acyclovir</a> <a href=>buy diflucan online</a> <a href=>indocin generic</a> <a href=>tenormin 50 mg</a>

162. AlfredKer -- 2017-07-21 03:21

wh0cd75507 <a href=>amantadine</a> <a href=>eldepryl</a> <a href=>reglan for lactation</a> <a href=>entocort ec price</a> <a href=>indinavir without prescription</a> <a href=>hytrin for bph</a> <a href=>imdur</a> <a href=>femara online</a>

163. BennySedly -- 2017-07-21 07:32

wh0cd985820 <a href=>lamictal</a> <a href=>capoten</a> <a href=>styplon</a> <a href=>amaryl</a>

165. CharlesTew -- 2017-07-22 02:22

wh0cd883731 <a href=>astelin nasal spray</a> <a href=>quibron-t</a> <a href=>shatavari</a> <a href=>pilex</a> <a href=>hytrin</a> <a href=>benadryl</a> <a href=>shatavari</a>

166. KennethCeali -- 2017-07-22 04:54

wh0cd561205 <a href=>buy actos online</a> <a href=>namenda 10 mg tablet</a> <a href=>revatio</a> <a href=>zyrtec</a> <a href=>zyvox</a> <a href=>artane</a> <a href=>buy prilosec</a> <a href=>buy alesse</a>

167. CharlesTew -- 2017-07-22 06:07

wh0cd46883 <a href=>phenergan</a>

169. CharlesTew -- 2017-07-22 11:02

wh0cd340779 <a href=>citalopram</a> <a href=>zithromax</a> <a href=>serophene</a> <a href=>bentyl</a>

172. CharlesTew -- 2017-07-23 16:01

wh0cd606060 <a href=>propranolol</a> <a href=>get more info</a> <a href=>atarax</a> <a href=>alavert</a>

174. TracySueme -- 2017-07-24 03:40

wh0cd643235 <a href=>altace</a> <a href=>celebrex</a> <a href=>metformin no prescription</a> <a href=>paxil</a>

175. CharlesTew -- 2017-07-25 00:03

wh0cd503957 <a href=>propecia</a> <a href=>diclofenac 75mg dr tab</a> <a href=>cheap crestor</a> <a href=>zetia prices</a>

176. CharlesTew -- 2017-07-25 08:01

wh0cd254911 <a href=>synthroid pills</a>

179. CharlesTew -- 2017-07-25 15:00

wh0cd181444 <a href=>altace</a> <a href=>eulexin without prescription</a> <a href=>zyvox iv cost</a> <a href=>serevent diskus</a>

180. Michaeltup -- 2017-07-25 18:30

wh0cd26818 <a href=>buy furosemide online</a> <a href=>buy bupropion</a> <a href=>metformin without prescription</a>

182. BennySedly -- 2017-07-26 06:24

wh0cd152820 <a href=>cafergot</a> <a href=>buspar online</a> <a href=>Cymbalta</a>

184. CharlesTew -- 2017-07-26 18:21

wh0cd299762 <a href=>generic for lipitor</a> <a href=>diflucan prescription</a> <a href=>Triamterene Hctz 75 Mg</a>

187. KennethCeali -- 2017-07-27 22:21

wh0cd462943 <a href=>generic micronase</a> <a href=>remeron</a> <a href=>buy isoptin</a> <a href=>imodium</a>

188. KennethCeali -- 2017-07-28 02:02

wh0cd609891 <a href=>HYDROCHLOROTHIAZIDE ONLINE</a> <a href=>synthroid lowest prices</a> <a href=>buy diflucan</a>

189. Eugenegussy -- 2017-07-30 09:52

wh0cd189996 <a href=>check this out</a> <a href=>atarax tablets</a> <a href=>medrol</a> <a href=>plavix without rx</a>

192. KennethCeali -- 2017-07-30 12:03

wh0cd303600 <a href=>buy diclofenac</a> <a href=>BUY MOTILIUM</a> <a href=>article source</a>

193. BennySedly -- 2017-07-30 12:46

wh0cd699593 <a href=>metformin 500 mg tablets</a> <a href=>buy crestor</a> <a href=>prozac uk</a>

194. CharlesTew -- 2017-07-30 18:31

wh0cd9704 <a href=>flagyl pills</a> <a href=>Benadryl</a>

195. TracySueme -- 2017-07-31 07:21

wh0cd308326 <a href=>where to buy cymbalta</a> <a href=>acyclovir 800 mg</a> <a href=>zithromax antibiotic</a> <a href=>generic medrol</a>

196. CharlesTew -- 2017-07-31 08:22

wh0cd744444 <a href=>zestril</a> <a href=>entocort</a> <a href=>actos</a> <a href=>menosan</a> <a href=>generic bystolic alternative</a>

197. Eugenegussy -- 2017-07-31 09:32

wh0cd381790 <a href=>bentyl</a> <a href=>buy prednisolone</a> <a href=>adalat tabs</a>

198. KennethCeali -- 2017-07-31 10:23

wh0cd715814 <a href=>Elavil 10 mg</a>

200. KennethCeali -- 2017-07-31 14:01

wh0cd862762 <a href=>hair loss cream</a> <a href=>tegretol</a> <a href=>cheap plan b decks</a> <a href=>astelin nasal spray otc</a> <a href=>vitamin c iv cost</a> <a href=>coreg cr 10mg</a>

201. anonymous -- 2018-01-07 17:34
202. anonymous -- 2018-01-07 17:35
203. James Billy -- 2018-03-05 07:41

You are the Savior! A few days later I was looking for a software network - many articles and lots of discussions - but many have no ideas! and I am Best Assignment Writer who offers academic assistance at all levels.

204. -- 2018-04-28 15:35

Unbelievable and interesting article. You always have good things to share with us. Thank you for continuing to write these posts.

Click here : web 2.0 profile backlinks

205. anonymous -- 2018-05-03 14:23

Other than this, students feel hard this task as selection of topics is the vital step, but involves numerous phases of refinement. Business Assignment Assistance

206. Victor Baily -- 2018-05-28 08:34

This Site And The Resources You Provide Is Really Nice Keep It Up. Assignment Help

207. Melodiec Bredahu -- 2018-05-28 11:43

John Arnold Is An Academic Writer Of The Dissertation-Guidance. Who Writes Quality Academic Papers For Students To Help Them In Accomplishing Their Goals. Online STATA Project Help

208. -- 2018-05-29 08:09

Visit our website to know the reviews of different assignment writing service providers. <a href="">allassignmenthelp reviews</a>

209. anonymous -- 2018-07-11 12:06

Thank you so much for this solution my friend is looking for this..really no doubt maths is very much difficult like I don't understand math so one day there is a product that i found from this site, on this product guidance page there is a graph of some approximation and believe me its out my mind.. that day I realise I much learn mathematics.

210. -- 2018-07-16 07:29

Thanks for sharing this solution because students <a href="">write my essay online</a>

211. essaymafia -- 2018-07-19 09:59

It is additionally imperative to check regardless of whether the mentor to you has incredible relational abilities. School educational programs are intended to be educative, however, is not really so for kids who bother to keep up the pace with their school and see all that is being instructed.... A noteworthy explanation behind this is the quick pace of giving exercises and in addition shameful correspondence with respect to the instructors. Discover at from others whether or the guide can give the exercises to your youngsters in an unmistakable way.

212. Stella -- 2018-07-20 14:56

Mothers (and Dads), my point in composing this is to give you some beginning stages. What's more, to tell you that you are not the only one! I know it can be disillusioning at first to take in your youngster is battling somehow. However, it can likewise feel like a weight has been lifted to know how your kid learns and that there are approaches to help and enable your little one. I know whether you are in a school setting, Dream Assignment Help UK you should disclose to your kid why they may go to an uncommon class or step through exams uniquely in contrast to alternate children. You need to confide in yourself to know how to converse with your youngster. There are books for kids that discussion about dyslexia and learning issues in a positive light, for example, Thank You, Mr. Falker by Patricia Polacco, The Alphabet War by Diane Robb, and for more seasoned kids May B via Caroline Rose or Niagra Falls, Or Does It? By Henry Winkler (yes, Fonzie from Happy Days!)Attempt to accentuate his/her qualities and affinities and don't just spotlight on his/her shortcomings and challenges. Remind your tyke that he/she can, to be sure, learn however that he/she learns exceptionally, and that is OK! We as a whole are remarkable and have our own particular qualities and shortcomings. Love your tyke for their identity and ideally, they will locate the correct devices to make learning take off!

213. riteaid survey -- 2018-08-09 16:38

Rite Aid Corporation is a drugstore chain in the United States. It is headquartered in Camp Hill, East Pennsboro Township, Cumberland County, Pennsylvania, near Harrisburg.

214. anonymous -- 2018-08-16 14:14

Thanks for sharing high value and informative article with us. And i hope this article is very helpful and useful for everyone.

215. My Assignment help -- 2018-08-18 07:06

Excellent information on your blog, thank you for taking the time to share with us. Amazing insight you have on this, it's nice to find a website the details so much information about different artists. visit here:-

216. anonymous -- 2018-08-18 07:11

Thank You So Much For providing the important information. I am Max Willor, SEO Expert in a reputed company. All of your information is very useful to me. I am providing Assignment Help Australia service to students.

217. dissertation help -- 2018-08-18 12:04

I am Ted Johnson, SEO Expert in a reputed company ukdissertationhelp .I would like to offer you guys tons of thanks for the detailed information and most importantly the information about the marketing strategies is top notch. visit here:-

218. Economics dissertation topics -- 2018-08-20 13:05

Students Assignment Help has the dedicated and expert team of professional writers who provide perfect economics dissertation topics. Our writing experts have complete knowledge of all academic disciplines to assist students. Our expert writers are highly experienced from the top universities. For more info visit

219. My Assignment Help -- 2018-08-22 08:26

This is an excellent post I seen thanks to share it. It is really what I wanted to see hope in future you will continue for sharing such a excellent post. Visit More:

220. mybkexperience -- 2018-08-27 07:06

Burger King (BK) is an American global restaurant chain of hamburger fast food. In the year 1953, Burger King was started in the name of Insta-Burger King in Jacksonville, Florida based chain restaurant.

221. dissertation help ireland -- 2018-08-27 08:59

Great Article it its really informative and innovative keep us posted with new updates. It was really valuable. Thanks a lot.

222. anonymous -- 2018-08-28 09:12

Awesome article keep doing this check out my website

223. anonymous -- 2018-08-30 09:53

Pretty! This was an incredibly wonderful article. Thanks for providing this information.

224. anonymous -- 2018-08-30 10:26

Are you looking for Dissertation Writing Services? is providing dissertation writing service company in the UK from the qualified expert. Our 3000+ dissertation writing experts are available 24x7 to help the UK students and bring forth the peerless dissertations end of the day without asking an amount that exceeds the budget.

225. Engineering Project Help -- 2018-09-01 07:41

Thanx for sharing such useful post keep it up :)

226. content Writing Help -- 2018-09-01 10:47

Things are very open and intensely clear explanation of issues. was truly information. Your website is very beneficial.

227. Students Assignment Help Singapore -- 2018-09-01 12:29

A student in the Singapore and want a best professional assistance then simply visit at and take our Students Assignment Help Singapore services at a reasonable cost. Our top academic experts will guide you towards finishing your tricky assignments easily.

228. Rafael Turnbull -- 2018-09-03 14:45

We Also Share Some Information About Our Business Case Study Writing Service

229. Johnny Korey -- 2018-09-04 14:19

Thanx For Sharing Such Useful Post Keep It Up :) Java Homework Help

230. Assignment Help -- 2018-09-15 09:05

The above post has given reliable and genuine information about Assignment Help Australia. Looking forward to avail their eminent services.

232. Elena -- 2018-09-17 17:35

Great information, it's good to read such things. Elena

233. anonymous -- 2018-10-03 20:53

World Series 2018 Live watch World Series 2018 Live how to watch World Series 2018

<a href="">World Series 2018 Live</a>

234. anonymous -- 2018-10-04 11:43

For students help you just look at the Assignment help given by the assignment experts. Opting for our Online Assignment Help could be ideal for such students and they can get a complete assignment solution n from us.

235. anonymous -- 2018-10-09 18:54

Your website is really cool and this is a great inspiring article. <a href=''>3rd eye activation</a>

236. anonymous -- 2018-10-12 09:21

Thank you for posting such a great blog! I found your website perfect for my needs. It contains wonderful and helpful posts. Keep up the good work. Thank you for this wonderful Blog! Visit Us:

237. anonymous -- 2018-10-12 13:53

Wonderful article, thanks for putting this together! This is obviously one great post. Thanks for the valuable information and insights you have so provided here.

238. anonymous -- 2018-10-14 10:06

This was a really great contest and hopefully I can attend the next one. It was alot of fun and I really enjoyed myself..

239. anonymous -- 2018-10-16 07:55

Our assignment helpers and expert academic writers in UK are available round the clock in order to be able to help college students and university students with various technicalities and aspects of writing. From helping students of UK colleges and universities with their term papers, to coursework and dissertations to assisting others with homework-based assignments, all our writers are here to help you out. If you ever need assignment help in UK, feel free to give us a call or email your assignment details.

uk assignment help

240. anonymous -- 2018-10-16 19:35

This was an extremely incredible challenge and ideally I can go to the following one. It was alot of fun and I truly lived it up.

241. anonymous -- 2018-10-18 10:14

I have used feed-forward Artificial Neural Networks to approximate any continuous function and found to be the best Custom Printed Boxes

242. anonymous -- 2018-10-21 07:03

I think this is a really good article. You make this information interesting and engaging. You give readers a lot to think about and I appreciate that kind of writing. IEPNW

243. anonymous -- 2018-10-22 11:35

I really loved reading your blog. It was very well authored and easy to undertand. Unlike additional blogs I have read which are really not tht good. I also found your posts very interesting. In fact after reading, I had to go show it to my friend and he ejoyed it as well! Ado Berlin

244. -- 2018-10-29 13:13

Great post! I am really getting ready to across this information, it is very helpful my friend. Also great blog here with all of the valuable information you have. Keep up the good work you are doing here. Avail No 1 Essay Writing Services from certified PhD writers. It is extremely helpful for me. You can email us at or Phone Number - 020 8144 9988. For More Details Visit :

245. Hannah Lewis -- 2018-10-31 06:31

Oh I hate sin cos tan so much! Those math should be taught in university only, for math enthusiastic people, not for me :( email qna thank you for sharing the interesting article.

246. -- 2018-10-31 12:30

All You Need To Know About CDR Reports As Engineering Students

247. -- 2018-10-31 16:06
248. anonymous -- 2018-11-01 10:18


Nice website.

Please visit our site: -

249. anonymous -- 2018-11-01 18:31
250. anonymous -- 2018-11-09 11:19

Great information, it's good to read such things.

251. -- 2018-11-24 06:00

I Personally Like Your Post, You Have Shared Good Article. It Will Help Me In Great Deal.

252. Assignment Hhelp Online -- 2018-11-24 06:55

I Am So Happy To Read This. This Is The Kind Of Manual That Needs To Be Given And Not The Random Misinformation That's At The Other Blogs.

253. Case Study Solution -- 2018-11-24 07:06

I'm Getting Excited About This Kind Of Beneficial Information Of Your Stuff In The Future

254. Case Study Solution -- 2018-11-24 07:16

Amazing Article Thanks Or Sharing..

255. Case Study Solution -- 2018-11-24 07:30

I Must Say We Should Have An Online Discussion On This.

256. Case Study Solution -- 2018-11-24 07:41

Amazing Article Thanks Or Sharing..

257. Case Solution -- 2018-11-24 07:56

By Visiting This Site I Found Cool Stuff Here Keep It Up.

258. Case Study Solution -- 2018-11-24 08:08

My Friend Recommended This Blog And He Was Totally Right Keep Up The Good Work

259. Admission Essay Writing Service -- 2018-11-24 08:20

John Arnold Is An Academic Writer Of The Dissertation-Guidance. Who Writes Quality Academic Papers For Students To Help Them In Accomplishing Their Goals.

260. Engineering Homework Help -- 2018-11-24 08:32

Only Professional Writers Can Make This Kind Of Material, Cheers

261. Best Essay Writing Service -- 2018-11-24 08:46

I'm Getting Excited About This Kind Of Beneficial Information Of Your Stuff In The Future

262. Do My Online Exam -- 2018-11-24 10:02

Good Way Of Telling, Good Post To Take Facts Regarding My Presentation Subject Matter, Which I Am Going To Deliver In My College

263. Assignment Help USA -- 2018-11-24 10:23

This Is Really Great Work. Thank You For Sharing Such A Useful Information Here In The Blog.

264. HR Homework Help -- 2018-11-24 10:37

We Also Share Some Information About Our Business

265. Online Job Placement Test -- 2018-11-24 10:51

Only Professional Writers Can Make This Kind Of Material, Cheers

266. Literature Review Writing Help -- 2018-11-24 11:14

Economicskey Provides Quality Economics Homework Help For Students.

267. Proposal Essay Topics Online Help -- 2018-11-24 11:28

By Visiting This Site I Found Cool Stuff Here Keep It Up.

268. Pay Me To Do My Online Class -- 2018-11-24 11:44

This Was A Great And Interesting Article To Read. I Have Really Enjoyed All Of This Very Cool Information

269. Pay Me To Do My Online CoursePay Me To Do My Online Course -- 2018-11-24 12:00

Amazing Article Thanks Or Sharing..

270. Write My Homework -- 2018-11-24 12:12

I’m Really Impressed With Your Article, Such Great & Usefull Knowledge You Mentioned Here

271. Assignment Help UK -- 2018-11-24 12:26

My Friend Recommended This Blog And He Was Totally Right Keep Up The Good Work

272. Best VPN Guru -- 2018-11-24 17:12

I hope such approximation functions help me in making my VPN solutions and predictions.....thanks for sharing

273. Dissertation Help -- 2018-11-27 12:32

Excellent and nice post. It will beneficial for everyone. Thanks for sharing such a wonderful post. Avail No 1 Dissertation Help UK from certified PhD writers. It is extremely helpful for me. You can email us at or Phone Number - 020 8144 9988.

Visit here:-

274. anonymous -- 2018-11-28 08:06

Thanks for sharing high value and informative article with us. And I hope you'll publish more article.

275. Thesis writing help UAE -- 2018-11-30 11:45

Students Assignment Help is catering its thesis writing help UAE services for all students. We provide our services without missing the deadline so that you can receive good marks in your academic career and can impress your professor. For more info visit

276. Assignment Help -- 2018-12-04 13:29

We offer Custom and Professional Assignment Writing Help to both College and University students who are stuck with their research projects, essays, case studies, business reports and need Urgent <a href =>Assignment Help</a>.

277. Adam Hebrew -- 2018-12-17 10:36

Our online assignment help and writing services have been devised keeping all your needs in mind. Through sheer hard work and consistency in our performance, we’ve earned the title of being the most popular assignment writing agency in the world. Get your assignment done Online assignment writing service offered by Uk professional assignment experts from UK.

278. My Assignment Help -- 2018-12-18 11:55

We at <a href =>My Assignment Help</a> offer all types of online academic services, be it homework help, coursework help, case study help, thesis/research paper writing help and online exam help.

279. anonymous -- 2018-12-22 08:43 have a pool of talented Online Assignment Helper that exactly know what students desire for. The writers in our team can easily get the best masters and Ph.D. degree from the prominent universities around the world. We will easily do all your task well on the asked deadline. For more information:

280. anonymous -- 2019-01-05 09:34

Losing sleep over unfinished essays but can't afford the assistance of costly essay writers? Wondering whether the online service providers will write my essay cheap in UK? Facing a deadline for finishing the essay? Well, your anxieties end here. We have with us a team of essay writing services for UK students who can solve all your writing issues so that you score high grades in your essay assignment. They will deliver the completed essay to you well before the deadline. With decades of experience in helping students, this team of professional essay typer best in UK.

281. anonymous -- 2019-01-08 07:48

Tim Parsons is on Facebook. Join Facebook to connect with Tim Parsons and others you may know. Facebook gives people the power to share and makes

Tim Parsons is on Facebook. Join Facebook to connect with Tim Parsons and others you may know. Facebook gives people the power to share and makes

282. anonymous -- 2019-01-08 11:39

Hello, I am Sophia Miller.

Get the best perdisco assignment help in Australia.

283. Brendan Greene -- 2019-01-12 14:11

We are an independent company who providing technical support for for Norton products.If you are Facing problems your Norton products or having any issues during the work do call us and our company will fix your problem and you need any suggestion regarding any Norton product feel-free to call us on our toll-free numbers for further information.

284. -- 2019-01-14 07:00

Custom Essay Help UK has the devoted and master group of expert authors who give impeccable financial matters thesis points. Our composition specialists have finish information of every scholastic control to help understudies. Our master essayists are exceptionally experienced from the best colleges. For more information visit :

285. John -- 2019-01-16 09:21

If you are looking for a perfect essay writing service and you are interested

to find out if the essay writing service will provide you assistance for all your project needs; stop searching the Google…just click on the Order icon to get the desired essay whether its related to sociology or history. Our writers are always there to help you… Don’t take the risk.. Find the very best in writing. Just Buy online.

286. Assignment Help -- 2019-01-16 11:47

Assignment Studio provides : Assignment Help Australia | Leadership Assignment | Case Study Help | MyAssignmenthelp | Finance Assignment Help | Online Assignment Help | Assignment Help Sydney <a href =>Assignment Help</a>

287. beatrix -- 2019-01-19 10:03

Are struggling with tiresome research paper writing task? Poorly written research assignments with little to say can put your career at stake. We have been providing UK based high-quality research papers. Our unparalleled online services encapsulate information gathering through different observational, investigative and experimental procedures using the best practices that primarily abide by the scientific method.

288. -- 2019-01-21 13:42

Instruction isn't just to scrutinize and compose. It is tied in with using preparing for their own positive conditions and to utilize the data for their improvement. Moreover one can lead their own reality without depending upon others. It isn't tied in with making people capable, a capability is out and out phenomenal preparing. A capability is expected for the ability to peruse and compose, while preparing is finding the clarification for everything and using the examining composing aptitudes to improve their lives.

289. anonymous -- 2019-01-22 12:27 Ver peliculas online Ver peliculas online Ver peliculas 2019 Ver peliculas Gratis Ver Peliculas Completas Online

290. anonymous -- 2019-01-22 12:27

<a href=""></a> <a href="">Ver peliculas online</a> <a href="">Ver peliculas online</a> <a href="">Ver peliculas 2019</a> <a href="">Ver peliculas Gratis</a> <a href="">Ver Peliculas Completas Online</a>

291. Assignment Help -- 2019-01-24 08:14

Assignment Studio provides : Assignment Help Australia | Leadership Assignment | Case Study Help | MyAssignmenthelp | Finance Assignment Help | Online Assignment Help | Assignment Help Sydney <a href =>My Assignment Help</a>

292. well -- 2019-01-25 09:34

Online Essay Writers UK has the best policy to keep all your conversation confidential. Individual working is the key to our success.Having a professional team of essay writing experts,

293. metro pcs bill pay -- 2019-01-26 07:54

In this article, we are going to discuss various topics about MetroPCS with some information and instructions.

294. johnk21 -- 2019-02-04 14:03

Hey, I was searching for some information about FLVTO but somehow cam on your website. After reading few of your articles i found that your writing and research is amazing. I will surely ask you to write something about FLVTO also. Thanks.

295. anonymous -- 2019-02-05 08:18

Try to add only relevant data and information. If you feel that you are stuck, you can take assignment help online to get a proper idea. There are several web services which can help you to find out a suitable solution to your problems. <a href="">Assignment help</a>

296. anonymous -- 2019-02-06 06:44

Pleasant and educational post. A debt of gratitude is in order for such a great post,loved the post, continue posting fascinating post.

297. anonymous -- 2019-02-06 06:44

A debt of gratitude is in order for shared great data of giving best data. its progressively valuable and increasingly supportive. incredible doing continue sharing

298. Assignment Help -- 2019-02-09 11:26

Assignment Studio Provides: Assignment Help Australia | Leadership Assignment | Case Study Help | MyAssignmenthelp | Finance Assignment Help | Online Assignment Help | Assignment Help Sydney Get Assignment Help from World’s No.1 Assignment Help Company <a href =>difference between research paper and thesis</a>

299. anonymous -- 2019-02-12 05:54

I think you are right to say so. Your share is great, I found a lot of things in your post.

300. anonymous -- 2019-02-12 11:29

Thank you for posting such a great article! I found your website perfect for my needs. It contains wonderful and helpful posts. Keep up the good work!. Thank you for this wonderful Article!

301. anonymous -- 2019-02-20 10:54

Thanks for giving that type of information. Really enjoyed this blog post. Really looking forward to reading more <a href="">contentsea!</a>

302. watson769 -- 2019-02-20 13:04

<a href="">Performing Team Development Model</a>, and so do the other people. This region determines the amount of co-operation and communication I have with other people. When I first meet a new person, I don’t gel up quickly. I take a lot of time to get comfortable with the person and create the comfort zone. I don’t get too talkative or expressive in the beginning.

303. anonymous -- 2019-02-21 12:13

<a href="">Strategic National Australian Bank</a> has been referred to as one of the leading financial institution with its operations in New Zealand (NZ), Asia & Australia. NAB has been referred to as one of the leading financial institution in terms of number of customers and market capitalization.

304. watson769 -- 2019-02-22 12:03

<a href=" ">Safety Management Assignment</a> Act, 1995 requires all organizations to carry out a risk assessment before making an intervention. Risk assessment requires the risk to health and safety to be controlled so far as is reasonably practical.

Add New Comment