Q4. Floating-point numbers and arithmetic (a) For a B-based floating-point number system of p digits of precision, with exponent ranging from L to U, find the total number of normalized floating-point number in terms of 3, p, L, U. (b) Let a be a nonzero real number and fl(a) be a k-digit rounding approximation to a in base 2. Show that the relative error of round k-digits satisfy the bound - fl(a)| <2=k. a - a Hint: Look at the two cases of rounding separately. At least how many digits would one need in order to guarantee the rounding approximation fl(a) in binary has a relative error of at most 3 x 10-16? Comment on how this relates to the number of precision digits for double-precision floating-point numbers. (c) Recall from the IEEE 754 standard that fl(x) = x(1+ 8) with |8| < e for some machine precision e. Show that |FI(fl(x+ y) + z) – (x+ y + 2)| < (\x + y| + |x + y + zl)e + |x+ yle², |S(x + fl(y+ 2)) – (x+ y + z)| < (ly + 2| + |x + y + z[)e + \y + z|e². Hint: Let fl(x + y) = (x + y)(1+ d1) and expand out fl(fl(x+y) + z) = (fl(x+ y) + z)(1+ d2) = (d) Conclude from part (c) that floating-point addition is in general not associative, i.e. =.... fl(fl(r+ y) + 2) # fl(x+ fl(y+ z)). If |r+y| < |y+ z|, which order of summation from part (c) will give a smaller bound for the absolute error?
Q4. Floating-point numbers and arithmetic (a) For a B-based floating-point number system of p digits of precision, with exponent ranging from L to U, find the total number of normalized floating-point number in terms of 3, p, L, U. (b) Let a be a nonzero real number and fl(a) be a k-digit rounding approximation to a in base 2. Show that the relative error of round k-digits satisfy the bound - fl(a)| <2=k. a - a Hint: Look at the two cases of rounding separately. At least how many digits would one need in order to guarantee the rounding approximation fl(a) in binary has a relative error of at most 3 x 10-16? Comment on how this relates to the number of precision digits for double-precision floating-point numbers. (c) Recall from the IEEE 754 standard that fl(x) = x(1+ 8) with |8| < e for some machine precision e. Show that |FI(fl(x+ y) + z) – (x+ y + 2)| < (\x + y| + |x + y + zl)e + |x+ yle², |S(x + fl(y+ 2)) – (x+ y + z)| < (ly + 2| + |x + y + z[)e + \y + z|e². Hint: Let fl(x + y) = (x + y)(1+ d1) and expand out fl(fl(x+y) + z) = (fl(x+ y) + z)(1+ d2) = (d) Conclude from part (c) that floating-point addition is in general not associative, i.e. =.... fl(fl(r+ y) + 2) # fl(x+ fl(y+ z)). If |r+y| < |y+ z|, which order of summation from part (c) will give a smaller bound for the absolute error?
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solve question (c) (d)
![Q4. Floating-point numbers and arithmetic
(a) For a B-based floating-point number system of p digits of precision, with exponent ranging from L to
U, find the total number of normalized floating-point number in terms of 3, p, L, U.
(b) Let a be a nonzero real number and fl(a) be a k-digit rounding approximation to a in base 2.
Show that the relative error of round k-digits satisfy the bound
- fl(a)|
a -
a
Hint: Look at the two cases of rounding separately.
At least how many digits would one need in order to guarantee the rounding approximation fl(a) in binary
has a relative error of at most 3 x 10-16? Comment on how this relates to the number of precision digits
for double-precision floating-point numbers.
(c) Recall from the IEEE 754 standard that fl(x) = x(1+ d) with |8| < e for some machine precision e.
Show that
|FI(fl(x+ y) + z) – (x+ y + 2)| < (\x + y] + |x + y + z\)e + |x+ y|e²,
|F(x + fl(y+ 2)) – (x+ y + z)| < (ly + 2| + |x + y + zl)e + \y + 2|e².
Hint: Let fl(x + y) = (x + y)(1+ d1) and expand out fl(fl(x+y) + z) = (fl(x+ y) + z)(1+ d2) =
(d) Conclude from part (c) that floating-point addition is in general not associative, i.e.
=....
fl(fl(x + y) + 2) # fl(x+ fl(y+ z)).
If |r+y| < |y+ z|, which order of summation from part (c) will give a smaller bound for the absolute error?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d32e0fa-456e-4142-9cd1-f7ab868e09af%2F607aea0b-f1c3-4ebe-a58f-d9c0d7813091%2F1b6u7d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q4. Floating-point numbers and arithmetic
(a) For a B-based floating-point number system of p digits of precision, with exponent ranging from L to
U, find the total number of normalized floating-point number in terms of 3, p, L, U.
(b) Let a be a nonzero real number and fl(a) be a k-digit rounding approximation to a in base 2.
Show that the relative error of round k-digits satisfy the bound
- fl(a)|
a -
a
Hint: Look at the two cases of rounding separately.
At least how many digits would one need in order to guarantee the rounding approximation fl(a) in binary
has a relative error of at most 3 x 10-16? Comment on how this relates to the number of precision digits
for double-precision floating-point numbers.
(c) Recall from the IEEE 754 standard that fl(x) = x(1+ d) with |8| < e for some machine precision e.
Show that
|FI(fl(x+ y) + z) – (x+ y + 2)| < (\x + y] + |x + y + z\)e + |x+ y|e²,
|F(x + fl(y+ 2)) – (x+ y + z)| < (ly + 2| + |x + y + zl)e + \y + 2|e².
Hint: Let fl(x + y) = (x + y)(1+ d1) and expand out fl(fl(x+y) + z) = (fl(x+ y) + z)(1+ d2) =
(d) Conclude from part (c) that floating-point addition is in general not associative, i.e.
=....
fl(fl(x + y) + 2) # fl(x+ fl(y+ z)).
If |r+y| < |y+ z|, which order of summation from part (c) will give a smaller bound for the absolute error?
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