square root formulas

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I have developed formulas to calculate the square root of real numbers discovered from a pattern you can read details at :

Edited by mones

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The linking is too convoluted.  Can you supply an example directly?

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Quote
3.2. High precision example
let us choose n=18 and calculate2 using formula (5) with high precision program [9]:

2 = tan (asin( 2 / (2 + 0.5. 10-36 )) / 1018

The result is :
1.4142135623730950488016887242096980784812835277286296
2 n = 36 correct decimals

Why do you think that doing tangent and arc sinus, will be faster than other, already present, methods... ?

The methods to calculate square root, you can find on this website:

Your equations are missing parenthesis.. everywhere..

After entering your above equation to calculator, or wolfram alpha, we see result:

1.570796326794896618524215... × 10^-18

1.41421

Edited by Sensei

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You are new here and you obviously haven't read the rules like so many other don't.

But I will ask the moderators to explain the rules.

1)

Not all methods of root extraction require an initial 'guess'.

There is a perfectly satisfactory rote method a bit like long division, though admittedly it is multi step as is long division.
You would probably need to look up some algebra texts from 1890 to 1920 to find it though.

2)

Your methods claim a single step, but appear to require an inexhaustible supply of more advanced functions such as trigonometric ones to achieve this.
How many steps does the computer or calculator execute to obtain these, and what would do without them?

What about the accuracy obtainable for these?
For instance the tangent changes very rapidly indeed near its singularities.

Suppose you wanted the square root of the difference of two similar numbers with one known to a large number of decimal digits, so the difference looses many decimals?

How would your formulae be successful then?

3)

Suppose the real number you wish to extract the root from is a surd.

How do you handle those?

Edited by studiot

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The method is sort of interesting, but useless.

41 minutes ago, studiot said:

You would probably need to look up some algebra texts from 1890 to 1920 to find it though.

Not quite that far. My dad knows how to do it and he was born in the 30s
It seems the manual method is still available.

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8 hours ago, mathematic said:

The linking is too convoluted.  Can you supply an example directly?

I have enabled the public profile at

If the link do not work properly let me know

I have made updates and clarifications and restored the parenthesis notation according to norm that do not have any influence or change on the formulas according to the scientific context.

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9 hours ago, studiot said:

You are new here and you obviously haven't read the rules like so many other don't.

But I will ask the moderators to explain the rules.

1)

Not all methods of root extraction require an initial 'guess'.

There is a perfectly satisfactory rote method a bit like long division, though admittedly it is multi step as is long division.
You would probably need to look up some algebra texts from 1890 to 1920 to find it though.

2)

Your methods claim a single step, but appear to require an inexhaustible supply of more advanced functions such as trigonometric ones to achieve this.
How many steps does the computer or calculator execute to obtain these, and what would do without them?

What about the accuracy obtainable for these?
For instance the tangent changes very rapidly indeed near its singularities.

Suppose you wanted the square root of the difference of two similar numbers with one known to a large number of decimal digits, so the difference looses many decimals?

How would your formulae be successful then?

3)

Suppose the real number you wish to extract the root from is a surd.

How do you handle those?

This is not a calculation method only it is a fundamental mathematical law and formulas that illustrate the incontestable relation between the square root and the trigonometric functions according to a specific pattern discovery and exact formulas, which is the sole pattern known for the square root according to my knowledge since the Babylonian. you may not consider it for square root calculation but it is not possible to ignore a scientific discovery and a fundamental mathematical law which is proved to perform unlimited mathematical operations.
You can still use the sophisticated program and methods to calculate the square root since the main purpose of the formula is not that

1)  I said "they requires an initial guess or starting value  and / OR many calculations steps or iterations"  "OR"   means the guess is not always needed since I have developed my own method  of square root calculation METHOD WITHOUT guess entitled
"square root and cubic root of any positive real number calculation methods without an initial guess or approximate value"
2) THIS IS NOT a METHOD it is a formula based on a pattern discovery with one step mean that you have just to replace the number to calculate the square root without iteration or multiple steps
The trigonometric function are certainly complicated to calculate using infinite series NOT STEPS, there is no known simpler method for the trigonometric function.
But you know mathematics is not calculation only, there is an idea behind the formulas which is to illustrate the incontestable relation between the square root and the trigonometric functions according to a specific pattern discovery

that is why I put the condition
m   is the number of digits of x (which include decimals)
n    m  to ensure the accuracy
and that is why I introduced The PRECISION FACTOR 10^n to emphasis the pattern and to ensure that you deal always with integers. but you can use any number with any number of decimals
No one or any computers in this planet can and will never calculate the infinite number of decimals and no one can use an infinite number
this is another idea behind my formulas
"the limited use of correct decimals is a calculation method adopted in all scientific and engineering fields for calculations"
You can use the number as a function as it is illustrated in square root replacement
3) That is why I said "the trigonometric function should be calculated with high precision with  higher correct decimals calculation precision" using a high precision program and "the numerical accuracy of the functions "asin" near −π/2 and π/2 and "acos" near 0 and π is ill-conditioned and will thus calculate the angle with reduced accuracy in a computer implementation (due to the limited number of decimals)"
I suggest that you work on the advantages of the formulas and explore their applications since it prove the relation between the square root and the trigonometric function according to a specific pattern and law, which do not depend on me or any one, this is HOW IT WORKS. the formulas of nature are very powerful you can not change it or complain about it. but you can critic and discard any calculation METHOD including mine.

8 hours ago, John Cuthber said:

The method is sort of interesting, but useless.

Not quite that far. My dad knows how to do it and he was born in the 30s
It seems the manual method is still available.

JARGON

10 hours ago, Sensei said:

Why do you think that doing tangent and arc sinus, will be faster than other, already present, methods... ?

The methods to calculate square root, you can find on this website:

Your equations are missing parenthesis.. everywhere..

After entering your above equation to calculator, or wolfram alpha, we see result:

1.570796326794896618524215... × 10^-18

1.41421

This is not a calculation method only it is a fundamental mathematical law and formulas that illustrate the incontestable relation between the square root and the trigonometric functions according to a specific pattern discovery and exact formulas, which is the sole pattern known for the square root according to my knowledge since the Babylonian. you may not consider it for square root calculation but it is not possible to ignore a scientific discovery and a fundamental mathematical law which is proved to perform unlimited mathematical operations.
Your suggest method references is INCLUDED in my references
I have restored the parenthesis notation according to norm that do not have any influence or change on the formulas according to the scientific context.

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10 hours ago, John Cuthber said:

The method is sort of interesting, but useless.

Not quite that far. My dad knows how to do it and he was born in the 30s
It seems the manual method is still available.

Thank you for the link, John.  +1

2 hours ago, mones said:
Quote

Studiot:Suppose the real number you wish to extract the root from is a surd.

How do you handle those?

This is not a calculation method only it is a fundamental mathematical law and formulas that illustrate the incontestable relation between the square root and the trigonometric functions according to a specific pattern discovery and exact formulas, which is the sole pattern known for the square root according to my knowledge since the Babylonian. you may not consider it for square root calculation but it is not possible to ignore a scientific discovery and a fundamental mathematical law which is proved to perform unlimited mathematical operations.
You can still use the sophisticated program and methods to calculate the square root since the main purpose of the formula is not that

How about a simple one eg

$2\sqrt {\sqrt 3 }$

2 hours ago, mones said:

Why should anyone read it, whilst you choose not to read the forum rules?

Edited by studiot

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3 hours ago, mones said:

I have enabled the public profile at

If the link do not work properly let me know

!

Moderator Note

You need to present your work here. Not provide a link to somewhere else. (Especially when that link does not go to an explanation.) You can use references to offsite documents to provide extra information, but the rules of the forum require you to present your argument in this thread.

3 hours ago, mones said:

!

Moderator Note

You cannot decide who can respond or how. If you think a reply is inappropriate, use the report function.

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2 hours ago, mones said:

JARGON

Which words didn't you understand?

Most of the words are common enough

This might trouble some people

To write off the method as "jargon" seems a bit silly. My dad learned it as a schoolkid- so  did the rest of his classmates.

It has the enormous advantage over your method that you only need a pen and paper.
Your method requires an infinite set of trigonometric tables and is thus not any practical use.

Edited by John Cuthber

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I would like to make it clear that the subject itself ((square ) roots) has been of great interest and importance for thousands of years and remains so today.

So I wish to clearly separate discussion about the subject from discussion about some associated claims you have made which are arguable.

Yes the babylonians are the first recorded people to have considered this

Fast forward to ultra modern times to look at it from a totally different point of view.

3 hours ago, mones said:

This is not a calculation method only it is a fundamental mathematical law and formulas that illustrate the incontestable relation between the square root and the trigonometric functions according to a specific pattern discovery and exact formulas

You seem to be implying that you are offering not a calculation of the numerical value of a square root, but an exact replacement for the conventional square root when it appears in a formula, and further claim it is somehow simpler than just leaving it as a square root.

In what way is this better?

We often leave irrational numbers that are infinite decimals if expressed in decimal form, as just that. For instance    $\pi$.

Similarly we leave    $\sqrt 3$ just as it is to avoid an infinite decimal and make it exact.

What could be simpler?

You have also indicated that you think series methods of obtaining trigonometric values are the only ones available.

There are others.

In fact there are proceedures for constructing exact tables starting with trig functions and angles that have exact values and using trig transformation formulae for sum and difference and multiple angles.

You would find these in Hobson for instance.

Finally, if you are allowed one press of a button to obtain these trig values, why is that preferable to one press of a button to obtain the square root directly from the same calculator?

After all, that calculator can never be more accurate than its digits and algorithms.
The more algorithms you employ the greater the compounding or build up of error.

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3 hours ago, studiot said:

Thank you for the link, John.  +1

How about a simple one eg

23

Why should anyone read it, whilst you choose not to read the forum rules?

I did not read the forum rules it was not a choice I was in hurry, but I respect the rules and I expected only scientific opinion since this is a respectful science forum

As for surds

As I told you use the square root function replacement described in paragraph 3.4.

So the general formula for surds is

tan (asin((tan (asin(( x / (x + 0.5. 10-2n ))))) / 10n ) / ((tan (asin( x / (x + 0.5. 10-2n ))) / 10n)  + 0.5. 10-2n ))) / 10n

for 2sqrt(sqrt(3))

choose for example n=18 the correct number of decimals is 2n=36 and replace

2tan (asin((tan (asin(( 3 / (3 + 0.5 10^(-36) )))) / 10^18 ) / ((tan (asin( 3 / (3 + 0.5 10^(-36) ))) / 10^18)  + 0.5 10^(-36) ))) / 10^18

Test it on

and you get

the results with 36 correct decimals:

2.632148025904984921638437803593998110075341841792392364

2.632148025904984921638437803593998110320137180411644353

You can increase the correct decimals by just increasing n value

1 hour ago, studiot said:

I would like to make it clear that the subject itself ((square ) roots) has been of great interest and importance for thousands of years and remains so today.

So I wish to clearly separate discussion about the subject from discussion about some associated claims you have made which are arguable.

Yes the babylonians are the first recorded people to have considered this

Fast forward to ultra modern times to look at it from a totally different point of view.

You seem to be implying that you are offering not a calculation of the numerical value of a square root, but an exact replacement for the conventional square root when it appears in a formula, and further claim it is somehow simpler than just leaving it as a square root.

In what way is this better?

We often leave irrational numbers that are infinite decimals if expressed in decimal form, as just that. For instance    π .

Similarly we leave    3 just as it is to avoid an infinite decimal and make it exact.

What could be simpler?

You have also indicated that you think series methods of obtaining trigonometric values are the only ones available.

There are others.

In fact there are proceedures for constructing exact tables starting with trig functions and angles that have exact values and using trig transformation formulae for sum and difference and multiple angles.

You would find these in Hobson for instance.

Finally, if you are allowed one press of a button to obtain these trig values, why is that preferable to one press of a button to obtain the square root directly from the same calculator?

After all, that calculator can never be more accurate than its digits and algorithms.
The more algorithms you employ the greater the compounding or build up of error.

I am not imposing any thing to scientific community, I said this is what I found and this is the proof if you find a proof that is wrong simply discard it

I was talking about trigonometric calculation, table are not a method of calculation it is a pre-calculated  values stored in book or computer and I do not have any problem to use it

As I described for surds the formulas have the power for manipulation it can stand up alone without support, I say may be this is will be useful for science if not discard it

3 hours ago, John Cuthber said:

Which words didn't you understand?

Most of the words are common enough

This might trouble some people

To write off the method as "jargon" seems a bit silly. My dad learned it as a schoolkid- so  did the rest of his classmates.

It has the enormous advantage over your method that you only need a pen and paper.
Your method requires an infinite set of trigonometric tables and is thus not any practical use.

If your methods are better then simply discard my formula and do not use it, I did not understand the relation between method and formula which are completely different concept, any method can not stand up alone including mine they need always support but the formula have the power of nature it can stand up alone

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OK, without using a calculator (or computer), please calculate the square root of 123456 to 6 digits for me.

You will need to work out the tangents + arcsins etc by hand.

Let me know how you plan to do that.

Do you see why I don't think your method is useful?

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3 hours ago, mones said:

I am not imposing any thing to scientific community, I said this is what I found and this is the proof if you find a proof that is wrong simply discard it

I was talking about trigonometric calculation, table are not a method of calculation it is a pre-calculated  values stored in book or computer and I do not have any problem to use it

As I described for surds the formulas have the power for manipulation it can stand up alone without support, I say may be this is will be useful for science if not discard it

Perhaps there is a language difficulty?

I didn't say you are imposing on anybody.

And I didn't say your proposition is wrong.

I did say your proposition is interesting and about an important subject.

I did identify two separate aspects of it.

Firstly the actual calculation of a square root, correct to a specified number of digits.

Secondly the use of that formula in place the square root in some larger formula.

Finally I noted some of the things others have been looking at.
Are you not interested in seeing what others have done as well as telling them what you have done?

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13 hours ago, John Cuthber said:

OK, without using a calculator (or computer), please calculate the square root of 123456 to 6 digits for me.

You will need to work out the tangents + arcsins etc by hand.

Let me know how you plan to do that.

Do you see why I don't think your method is useful?

This is possible using Taylor series but  it will take a long time, again this is a formula not a method.

10 hours ago, studiot said:

Perhaps there is a language difficulty?

I didn't say you are imposing on anybody.

And I didn't say your proposition is wrong.

I did say your proposition is interesting and about an important subject.

I did identify two separate aspects of it.

Firstly the actual calculation of a square root, correct to a specified number of digits.

Secondly the use of that formula in place the square root in some larger formula.

Finally I noted some of the things others have been looking at.
Are you not interested in seeing what others have done as well as telling them what you have done?

I want every scientist to explore the advantages of my formula that is why I joined the forum, and this is exactly what I am looking for to identify all it's aspects and applications, and keep informing me on any new aspects or applications, it is very interesting to see what they will conclude because the square root has extremely   vast application every where in science, I want to emphasis that it is a formula using trigonometric function may be complicated but have an incredible flexibility, and this will not happen in few days you have all the time.

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55 minutes ago, mones said:

I want every scientist to explore the advantages of my formula that is why I joined the forum, and this is exactly what I am looking for to identify all it's aspects and applications, and keep informing me on any new aspects or applications, it is very interesting to see what they will conclude because the square root has extremely   vast application every where in science, I want to emphasis that it is a formula using trigonometric function may be complicated but have an incredible flexibility, and this will not happen in few days you have all the time.

What advantages.. ? Trigonometric functions are extremely slow to calculate (by CPU/FPU)...

sqrt(x) is just generalization of power of x^0.5

cube root(x) is just generalization of power of x^0.3333333(3)

ps. Make function that is calculating something faster than currently existing methods, and you will have attention from scientific community, and more importantly from computer designer community (faster operation is always welcome by programmers and users of applications). You need to benchmark currently existing methods and you own method to verify your method is faster than native, and alternative implementations.

Edited by Sensei

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7 hours ago, mones said:

This is possible using Taylor series but  it will take a long time

It would be much quicker to use a Taylor series to calculate the square root.

So, what's the point of your formula?

I cannot imagine any circumstance where replacing "sqrt(2)" by " tan (asin( 2 / (2 + 0.5. 10-36 )) / 1018"

or whatever would be an advantage (unless you were trying to hide the fact that you meant root 2.)

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I first encountered square roots in elementary school, where we taught a simple method without first guessing.

https://en.wikipedia.org/wiki/Methods_of_computing_square_roots  See digit by digit calculation.

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23 hours ago, John Cuthber said:

It would be much quicker to use a Taylor series to calculate the square root.

So, what's the point of your formula?

I cannot imagine any circumstance where replacing "sqrt(2)" by " tan (asin( 2 / (2 + 0.5. 10-36 )) / 1018"

or whatever would be an advantage (unless you were trying to hide the fact that you meant root 2.)

It mean that the square root can be expressed as  a Tangent of a specific angle and represented in a right triangle according to a specific pattern not arbitrarily, you can power the equation to enlarge the pattern to all number.

On ٩‏/١٢‏/٢٠١٨ at 9:00 AM, Sensei said:

What advantages.. ? Trigonometric functions are extremely slow to calculate (by CPU/FPU)...

sqrt(x) is just generalization of power of x^0.5

cube root(x) is just generalization of power of x^0.3333333(3)

ps. Make function that is calculating something faster than currently existing methods, and you will have attention from scientific community, and more importantly from computer designer community (faster operation is always welcome by programmers and users of applications). You need to benchmark currently existing methods and you own method to verify your method is faster than native, and alternative implementations.

Try this calculation method

for X= x/(x+0.510-2n)

You do not have to power X from the beginning each time use cumulative

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Using John Cuthber's example

sqrt(123456) = 351.3630601  directly on my 30 year old TI calculator.

If I choose another formula

sqrt(123456) = antilog(0.5*log(123456)) = 351.3630601

There is no perceptible difference in speed.

What keystrokes would you recommend to achieve this in your system?

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7 hours ago, mones said:

It mean that the square root can be expressed as  a Tangent of a specific angle and represented in a right triangle according to a specific pattern not arbitrarily, you can power the equation to enlarge the pattern to all number.

Well done.

You have rediscovered Pythagoras' theorem.

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10 hours ago, John Cuthber said:

Well done.

You have rediscovered Pythagoras' theorem.

Of course my formula PATTERN is deduced using Pythagorean theorem

17 hours ago, studiot said:

Using John Cuthber's example

sqrt(123456) = 351.3630601  directly on my 30 year old TI calculator.

If I choose another formula

sqrt(123456) = antilog(0.5*log(123456)) = 351.3630601

There is no perceptible difference in speed.

What keystrokes would you recommend to achieve this in your system?

If you mean just calculating using my formula on computer  scientific calculator choose Radians

calculate tan(antisin (123456/123456.0000005))/1000=351.3630601

when I tested your formula with log on my old computer it takes a tremendous time that I have to stop the operation without result.

I am not proficient in computer science or all the science field, that is why I asked scientist to find out if there is possible benefits, but if I can give my opinion I will not hesitate

My formula is like a tool you may not use it and you may need it,

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4 hours ago, mones said:

If you mean just calculating using my formula on computer  scientific calculator choose Radians

calculate tan(antisin (123456/123456.0000005))/1000=351.3630601

when I tested your formula with log on my old computer it takes a tremendous time that I have to stop the operation without result.

Why do you think using radians would make any difference?
It doesn't on mine and is slightly slower.

Both using DMS or rads is much slower than using the log / antilog formula, which is effectively instantaneous.

However

I cannot enter 123456.0000005 into my caclulator.
Not many  calculators have that many digits.

Mine displays 10.
I 'm not sure if it calculates to further guard digits, but I think so.

If I use 123456/123456.0005 then I get 11110.42436, not the right answer.

4 hours ago, mones said:

I am not proficient in computer science or all the science field, that is why I asked scientist to find out if there is possible benefits, but if I can give my opinion I will not hesitate

My formula is like a tool you may not use it and you may need it,

I am not trying to dismiss your formula, or discredit it. Who knows, there may one day be a use for it.  Knowledge should not be disgarded because it is currently unwanted.

I am taking you up on your request to examine its use in real situations and have offered you the results of my simple trials.

So if I was sitting on top of some mountain in Arabia (as I have been) and wanted a square root (which I have) but have not lugged a heavy computer up there (which I didn't) my thoughts are.

1) Modern (small) calculators have a square root button.

2) This is faster than trigonometric solutions.

3) If you, like me, did not have a calculator with even a square root button, let alone trig and log functions, you would have to rely on human ingenuity (as I did).

4) My companion took a whole afternoon to extract a single root, using a version of Newton's method on his calculator.

5) My digit bracketing method takes a few minutes per root extraction and automatically confirms the answer at the end of the extraction. Checking is important in real life.

Edited by studiot

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I feel the comments are hard here.

It's not every day that we read a new method to compute square roots. Maybe the proposal is not faster than the algorithms running presently in computer libraries, but it may have other benefits. At least, it opens news windows.

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19 hours ago, studiot said:

Why do you think using radians would make any difference?
It doesn't on mine and is slightly slower.

Both using DMS or rads is much slower than using the log / antilog formula, which is effectively instantaneous.

However

I cannot enter 123456.0000005 into my caclulator.
Not many  calculators have that many digits.

Mine displays 10.
I 'm not sure if it calculates to further guard digits, but I think so.

If I use 123456/123456.0005 then I get 11110.42436, not the right answer.

I am not trying to dismiss your formula, or discredit it. Who knows, there may one day be a use for it.  Knowledge should not be disgarded because it is currently unwanted.

I am taking you up on your request to examine its use in real situations and have offered you the results of my simple trials.

So if I was sitting on top of some mountain in Arabia (as I have been) and wanted a square root (which I have) but have not lugged a heavy computer up there (which I didn't) my thoughts are.

1) Modern (small) calculators have a square root button.

2) This is faster than trigonometric solutions.

3) If you, like me, did not have a calculator with even a square root button, let alone trig and log functions, you would have to rely on human ingenuity (as I did).

4) My companion took a whole afternoon to extract a single root, using a version of Newton's method on his calculator.

5) My digit bracketing method takes a few minutes per root extraction and automatically confirms the answer at the end of the extraction. Checking is important in real life.

the radians choice is imposed by the calculator you can choose degrees or grads it is the same,

Since it is 0.510^(-2n) you have to enter an even decimals number of "0"(2,4,6,8,10...) it is better to just add "0" decimals than to calculate 10^(-2n) and consuming time, it is the same result. in your example you entered 3 "0" which is incorrect.

I do not have facilities to benchmark a compared solution about square root calculation with my formula so I can not give a scientific reply, but you can calculate my formula with hand and trigonometric function by hands also using series expansion

the speed of calculation depend also on the quality of program and techniques and how it is performed

My formula is the result of hard work and high quality research with deep analysis resolving a very tough problem since the Babylonian, the speed of calculation is just an aspect of the formula that I wish to be the fastest formula, but I am not a computer scientist to work on this issue.

5 hours ago, Enthalpy said:

I feel the comments are hard here.

It's not every day that we read a new method to compute square roots. Maybe the proposal is not faster than the algorithms running presently in computer libraries, but it may have other benefits. At least, it opens news windows.

This is what I am trying to say; the formula may have many aspects and applications since the square root is used every where in science for many purposes,