For each problem, briefly explain/justify how you obtained your answer. This will help us de- termine...

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For each problem, briefly explain/justify how you obtained your answer. This will help us de- termine your understanding of the problem and whether or not you got the correct answer. More- over, in the event of an incorrect answer, we can still try to give you partial credit based on the explanation you provide. Again, please type your solution and submit it on Gradescope! 1. (15 points) Represent each of the following using the 8-bit floating-point format (which had 3 bits for the mantissa and 4 bits for the excess-7 exponent). Show your intermediate work. (a) 3.25 (b) -81.5 (c) 1/32 2. (10 points) Consider the 8-bit floating-point format, including 3 bits for the mantissa and 4 bits for the excess-7 exponent. Show your intermediate work. (a) What 8-bit pattern represents the number -0.625= ? (b) What base-10 integer or fraction does 01001001 represent? 3. (15 points) Consider a 7-bit floating-point representation with 3 bits for the excess-3 expo- nent and 3 bits for the mantissa. (a) How would 0.275 (10) be represented in this 7-bit representation? (b) What decimal value does 0100110 represent? (c) What decimal value does 1001100 represent? 4. (10 points) To store z = (do.didi dididi+1) XBe, where 1 do 9,0 d; 9, i=1,t-1, and 3 Z22 is the base, using only t digits, it is possible to use one of the following two strategies. Chopping: ignore the digits ded+1, and . ... f(x) = do.dd-1 x Be (1) 4. (10 points) To store z = (do.dida didide+1) X B, where 1 do 9,0 d 9, i=1,-1, and Z22 is the base, using only t digits, it is possible to use one of the following two strategies. Chopping: ignore the digits ded+1, and . . Rounding: consult de to determine the approximation Edo.dd-1x B (do.dd-1 +8-) x B di f(x) = 120 animal fl(x)=do.ddt-1 X Be function (5) Let z f(x) = g x 3, where a 0 and g is the normalized, signed mantissa. Prove that the absolute error committed in using the floating point representation of z is bounded by -f(x)| (Note: Please demonstrate your logic in detail. Your proof should hold for any t, d,, and 3. Proof by providing an example is considered invalid and viewed as no attempt/blank solution!!!) (1) if fl is implemented using chopping 1) if fl is implemented using rounding 2) (2) (1 mofr) can he transformed into another 5. (30 points) The function f(x, 6) = cos(x + 5) - cos(x) can be transformed into another form, f2(x, 6), using the trigonometric formula. cos (4) cos() = -2 sin 4+1 2 sin Thus, fi and f2 have the same values, in exact arithmetic, for any given argument values and S. (3) (a) Derive f2(x, 8) using equation 3. Show that, analytically, f(a) or fa(8) are effective approximations of the function - sin(x) for & sufficiently small. (b) Verify correctness of the equation 3. (c) Calculate the g(x, 6) = f(6) + sin(x) and 92(x, 8) = f(26) + sin(x) for x = 3 and 6 = 1e 11. Which approximation of sin(x) is better? Explain why you think it is better analytically. (a) 15.0625 (b) -35.6875 (c) 86.5625 (Note: Again, please demonstrate your logic in detail. Your proof should hold universally. Proof by providing an example is considered invalid and viewed as no attempt/blank solution!!!) 2 6. (15 points) Represent the following three decimal numbers as single precision float. Show: a scientific notation representation the significand, exponent, and sign bits (32 bits total) a hexadecimal representation of these 32 bits . 7. (25 points) For the following functions, compute the relative condition numbers. Are the functions ill-conditioned at any points (answer based on the value of the relative condition number)? (a) f(x)=x+c (b) f(x) = cr (c) f(x) = r (d) f(x) = e (e) f(x) = sin(x) (f) f(x) =tan(x) (g) f(x) = log(x) (h) f(x)=(1+x) For each problem, briefly explain/justify how you obtained your answer. This will help us de- termine your understanding of the problem and whether or not you got the correct answer. More- over, in the event of an incorrect answer, we can still try to give you partial credit based on the explanation you provide. Again, please type your solution and submit it on Gradescope! 1. (15 points) Represent each of the following using the 8-bit floating-point format (which had 3 bits for the mantissa and 4 bits for the excess-7 exponent). Show your intermediate work. (a) 3.25 (b) -81.5 (c) 1/32 2. (10 points) Consider the 8-bit floating-point format, including 3 bits for the mantissa and 4 bits for the excess-7 exponent. Show your intermediate work. (a) What 8-bit pattern represents the number -0.625= ? (b) What base-10 integer or fraction does 01001001 represent? 3. (15 points) Consider a 7-bit floating-point representation with 3 bits for the excess-3 expo- nent and 3 bits for the mantissa. (a) How would 0.275 (10) be represented in this 7-bit representation? (b) What decimal value does 0100110 represent? (c) What decimal value does 1001100 represent? 4. (10 points) To store z = (do.didi dididi+1) XBe, where 1 do 9,0 d; 9, i=1,t-1, and 3 Z22 is the base, using only t digits, it is possible to use one of the following two strategies. Chopping: ignore the digits ded+1, and . ... f(x) = do.dd-1 x Be (1) 4. (10 points) To store z = (do.dida didide+1) X B, where 1 do 9,0 d 9, i=1,-1, and Z22 is the base, using only t digits, it is possible to use one of the following two strategies. Chopping: ignore the digits ded+1, and . . Rounding: consult de to determine the approximation Edo.dd-1x B (do.dd-1 +8-) x B di f(x) = 120 animal fl(x)=do.ddt-1 X Be function (5) Let z f(x) = g x 3, where a 0 and g is the normalized, signed mantissa. Prove that the absolute error committed in using the floating point representation of z is bounded by -f(x)| (Note: Please demonstrate your logic in detail. Your proof should hold for any t, d,, and 3. Proof by providing an example is considered invalid and viewed as no attempt/blank solution!!!) (1) if fl is implemented using chopping 1) if fl is implemented using rounding 2) (2) (1 mofr) can he transformed into another 5. (30 points) The function f(x, 6) = cos(x + 5) - cos(x) can be transformed into another form, f2(x, 6), using the trigonometric formula. cos (4) cos() = -2 sin 4+1 2 sin Thus, fi and f2 have the same values, in exact arithmetic, for any given argument values and S. (3) (a) Derive f2(x, 8) using equation 3. Show that, analytically, f(a) or fa(8) are effective approximations of the function - sin(x) for & sufficiently small. (b) Verify correctness of the equation 3. (c) Calculate the g(x, 6) = f(6) + sin(x) and 92(x, 8) = f(26) + sin(x) for x = 3 and 6 = 1e 11. Which approximation of sin(x) is better? Explain why you think it is better analytically. (a) 15.0625 (b) -35.6875 (c) 86.5625 (Note: Again, please demonstrate your logic in detail. Your proof should hold universally. Proof by providing an example is considered invalid and viewed as no attempt/blank solution!!!) 2 6. (15 points) Represent the following three decimal numbers as single precision float. Show: a scientific notation representation the significand, exponent, and sign bits (32 bits total) a hexadecimal representation of these 32 bits . 7. (25 points) For the following functions, compute the relative condition numbers. Are the functions ill-conditioned at any points (answer based on the value of the relative condition number)? (a) f(x)=x+c (b) f(x) = cr (c) f(x) = r (d) f(x) = e (e) f(x) = sin(x) (f) f(x) =tan(x) (g) f(x) = log(x) (h) f(x)=(1+x)

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1 15 points a Representing 325 in the 8bit floatingpoint format Step 1 Convert 325 to binary 325 1101 Step 2 Normalize the binary representation 1101 1101 x 21 Step 3 Convert the normalized binary rep... View the full answer