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finite word length effects - related questions get little different results than expected - why?

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Seetha Rama Raju Sanapala 2017-12-22

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评论： Seetha Rama Raju Sanapala 2017-12-27

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We know that all computers suffer from finite word length effects. We can expect some errors - but why some related questions get different results for example - the following cut and paste from command window.

>> tand(90)
ans =
   Inf
>> tan(pi/2)
ans =
   1.6331e+16
>> cot(0)
ans =
   Inf
>> sin(pi)
ans =
   1.2246e-16
>> cos(pi/2)
ans =
   6.1232e-17

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Christoph F. 2017-12-22

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Integer numbers (like 0 and 90) can be represented accurately with the computers number formal, while irrational numbers cannot be represented without error.

John D'Errico 2017-12-22

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Even most simple decimals are not representable exactly. Thus 0.1 is NOT exactly represented as a floating point number. But all of this is just part of understanding how a computer stores numbers, and necessary for you to work with computers.

sprintf('%0.55f',0.1)
ans =
    '0.1000000000000000055511151231257827021181583404541015625'

Stephen23 2017-12-22

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@Seetha Rama Raju Sanapala: it is not clear what you are asking. If you are already aware that pi is not really equal to the mathematical constant, then surely you do not expect sin(pi) to be equal to zero?

Floating-point numbers have been explained a thousand times on this forum, but here are some good threads to read:

https://www.mathworks.com/matlabcentral/answers/57444-faq-why-is-0-3-0-2-0-1-not-equal-to-zero

https://www.mathworks.com/matlabcentral/answers/69-why-does-1-2-3-1-3-not-equal-zero#answer_271

https://www.mathworks.com/matlabcentral/answers/316889-why-two-equal-numbers-are-not-equal

https://www.mathworks.com/matlabcentral/answers/41536-why-is-2-24-10-22-4-not-equal-to-0

https://www.mathworks.com/matlabcentral/answers/20830-1-0010-does-not-equal-1-0010

This is well worth reading if you really want to understand how to work with floating point numbers:

https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html

Seetha Rama Raju Sanapala 2017-12-22

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I am aware of the finite word length effects and the limitations of the algorithms that are employed to get the answers (for example infinite series calculated with finite number of terms in practice and so on.). My question is really about why certain things you get correctly and the same things or closely related things you get with different accuracies. tan(pi/2) = cot(0). But you get two different answers from MATLAB. tan(90 degrees)=tan(pi/2 radians). But you get different answers in MATLAB. Does MATLAB not convert degrees into radians and evaluate?

Stephen23 2017-12-22

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编辑：Stephen23 2017-12-22

"But you get different answers in MATLAB. Does MATLAB not convert degrees into radians and evaluate?"

Clearly it does not, atleast not for those values. If you want to know the exact implementation then you will need to join TMW.

"My question is really about why certain things you get correctly and the same things or closely related things you get with different accuracies"

Because they are different algorithms using different input data. I see no reason why they should return the same output.

Walter Roberson 2017-12-22

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tand() and related routines special-case multiples of 90 degrees to get the exact answers

John D'Errico 2017-12-23

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You don't seem to understand that a number like pi is NOT represented exactly. It is just a number. And routines like TAND are generally very simple. They multiply your argument by pi/180, then call the regular tan function. You NEVER want to write code to compute a special function like the tangent twice. Then you will need to maintain two copies of the same code.

However, in the case of tand, it might even be smart enough to recognize some special inputs, like 90 degrees, then returning inf. cot will do the same. 0 is an easily recognized input. So cot special cases that one.

So what is pi in MATLAB? It is stored in binary form, as are all doubles, so only 52 bits of precision in the mantissa. The value used in MATLAB is:

sprintf('%0.55f',pi)
ans =
  '3.1415926535897931159979634685441851615905761718750000000'

However, the first 55 digits of pi are:

3.141592653589793238462643383279502884197169399375105821

As you can see, they stop looking the same after about the 16'th significant digit.

Seetha Rama Raju Sanapala 2017-12-27

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I know irrational numbers (including pi) decimal representation. My question is more of - If tand() and related routines special-case multiples of 90 degrees to get the exact answers, why can't these routines special case multiples of pi/2 radians to get the exact answers?

Walter Roberson 2017-12-27

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The radian-based trig functions are passed over to hardware instructions for performance reasons.

IEEE 754 defined accuracy requirements in terms of actual input bit patterns, and returning 0 in those cases would be inconsistent with the IEEE 754 accuracy requirements.

Seetha Rama Raju Sanapala 2017-12-27

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