GATE CSE 2004 | Question: 28

Question

What is the result of evaluating the following two expressions using three-digit floating point arithmetic with rounding?

$(113. + -111.) + 7.51$

$113. + (-111. + 7.51)$

• A. $9.51$ and $10.0$ respectively

• B. $10.0$ and $9.51$ respectively

• C. $9.51$ and $9.51$ respectively

• D. $10.0$ and $10.0$ respectively

Comments

Rewrite the smaller number such that its exponent matches with the exponent of the larger number.

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@amarVashishth 

Do you mean smaller number in terms of magnitude without the sign ?  As in the accepted answer −1.11×10^2+7.51×10^0   ,  -1.11 (also -111 < 7.51)  is smaller than 7.51, but we are rewriting 7.51 's exponent to match with 1.11' s exponent

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Comment, just see this comment and then attempt this question..

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using three-digit floating point arithmetic with rounding
so we cannot exceed 3 digits in any step , it needs to be rounded

Answer 1

$(113. + -111.) = 1.13 \times 10^2 + -1.11 \times 10^2 = 0.02 \times 10^2 = 2.0 \times 10^0$

$2.0 \times 10^0 + 7.51 \times 10^0 = 9.51 \times 10^0 $

$(-111. + 7.51) = -1.11 \times 10^2 + 7.51 \times 10^0 = -1.11\times 10^2 + 0.08 \times 10^2 = -1.03 \times 10^2 $

$113. + -1.03 \times 10^2 = 1.13 \times 10^2 + -1.03 \times 10^2 = 0.1 \times 10^2 = 10.0$

Reference: https://www.doc.ic.ac.uk/~eedwards/compsys/float/ 

Correct Answer: $A$

Comments

sir Is it not same thing as i had posted ?

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yes, I just gave an intermediate step :)

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Sir, your first explanation is nice , but for second ?
http://math.stackexchange.com/questions/1531692/two-expressions-using-three-digit-floating-point-arithmetic-with-rounding

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@Mithilesh I had told you not to edit others answer. Everyone won't be thinking like you.

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there is difference in the answers rt?

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@Arjun sir Your ref. link is as always amazing.

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+7.51∗10^0

why 7.51 is not rounded of to 8.00 in first part addition?

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Why it should be rounded? Rounding happens only when it is really required (as we loose precision) or when explicitly told. Also, this question is regarding fixed point arithmetic which is in GATE syllabus but not well covered in most books.

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@Arjun Sir,

Here in the following step:
(−111.+7.51) = −1.11∗10²+7.51∗10⁰ = −1.1∗102+0.08∗10²
The rule is to convert the exponent of smaller number as same as of the larger. But, here lager number is 7.51, whose exponent is zero. But, here you have converted the exponent of larger number to that of smaller's (i.e., exponent of 7.51 (larger) to that of -1.11 (smaller))?

Can you please explain it?

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So floating point additions are not associative right?

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Thanks for the link @Arjun sir thanks a lot ...really it was life saving ...

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@Arjun sir link is wonderful

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Thank you for explaination.

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In 3rd line of Arjun sir answer he wrote, −1.1×10^2+0.08×10^2=−1.03×10^2, but The actual result of this calculation is -1.02*10^2 ...

Which makes the answer as 11.0

Kindly explain If I am wrong here?

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The 1 missing was a typo. Fixed now.

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What is the significance of "three digit" floating point arithmetic? The floating point representation must have 3 digits in its mantissa? or floating point representation must have 2 digits in its mantissa with implicit digit just before decimal point?

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$7.51 * 10^{0}$ = $0.0751 * 10^{-2}$ will be rounded like $0.07 * 10^{-2}$

why 0.08 ?? [ rounding means tuncating to fit into the space]

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Reference link is not working!

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@arjun sir, link says that first add the mantissa then at last do the round off but in your steps you have taken round off first and then later done the addition of mantissa.

Though, the result won't change for this problem and I was crafting few other examples for which the answer was not changing. Not sure whether both will result the same ALWAYS.

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three-digit floating point arithmetic – means in total three digit before and after the decimal point? 

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@Arjun sir, In the link given it is said that the hidden bit is for binary floating-point numbers, but here the numbers are in decimal format. So why mantissa has only 2 digit?

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@Arjun Sir, @Kabir5454, I have the same question as @jatinmittal199510. 

EDIT: Ok, Got it. The next examples on that page shows that we first need to normalize the numbers and then start doing the operations.

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three-digit floating point arithmetic with rounding what does this mean ? we have to take total three digit before and after the decimal ?

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@Arjun sir reference link is awesome. 
Thanks

Answer 2

3 digit floating point arithmetic is used..
(113.+-111.)+7.51 = 2.00 + 7.51 = 9.51
113.+(-111.+7.51) = 113. + (-111. + 8.00) //rounding off to make compatible 7.51 and 111. with respect  3 digit floating point arithmetic
113. - 103. = 10.0

Comments

to apply operation ur operand should be THREE FLOATING POINT ARITHMATIC ..

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Please explain the meaning "three-digit floating point arithmetic with rounding"

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In n-digit floating-point arithmetic, numbers are represented by a signed n-digit integer and an exponent

113. in floating point is represented as 1.11 x 10^2

Answer 3

much better explanation

http://math.stackexchange.com/questions/1531692/two-expressions-using-three-digit-floating-point-arithmetic-with-rounding

Answer 4

2^nd part:  113. + (-111. + 7.51)

(-111. + 7.51) = - .111 x 10³ + .751 x 10

= 10( - .111 x 10² + .751)

= 10(-11.1 + .751)

= 10(- 10.349)

= - 103.49

now, 113. + (- 103.49) = .113 x 10³ - .10349 x 10³ 

= 10³(.113 - .103)  // ignor 49 bcz  three-digit floating point arithmetic with rounding

= 10³ x .010

= 10

Comments

I have a doubt why .10349 is rounded off to .103 its should be .104

please help

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because 0.103 is close to 0.10349… if it would have been 0.10351 then it would be 0.104