Question

2.4 Recall from class that MATLAB uses standard (IEEE) double-precision floating point notation: 52 bits 11 bits where each bit b Any Number- +/- (1.bbb...bbb)2 x 2 (bbb..bb2 102310 represents the digit 0 or 1. That is, the mantissa is always assumed to start with a 1, with 52 bits afterwards, and the exponent is an eleven bit integer (from 000..001 to 111...110) biased by subtracting 1023 Well, in "my college days" the standard was single-precision floating point notation in 32-bit words: 23 bits 8 bits Any Number- (1.bbb...bbb)2 x 2 (bbbbbbb, -12710' That is, the mantissa is always assumed to start with a 1, with 23 bits afterwards, and the exponent is an eight bit integer (from 00000001 to 11111110) biased by subtracting 127 to allow for an almost equal range of positive, zero, and negative exponents. And it still reserved all exponent bits identically equal to 00000000 for the number 0 (and "denormal" numbers), and 11111111 for ». a) Evaluate REALMAX (the largest possible positive number that is not infinity) for my (1990s) single- precision computer. 3 pts Express your value as a base 10 floating-point numbers with 3 sig. figs (e.g. 3.45 x 1025), and show your work! (What did you start with in binary, and how did you get that to decimal?) ii. Compare your value to today's double-precision computers (just type realmax in MATLAB) b) told you in class that machine precision in MATLAB, which is defined as the difference between 1 and the next largest storable number, is approximately 2.2204 x 10-16 in today's double-precision computers (just type eps in MATLAB to confirm) i. Evaluate machine precision for the single-precision computers from "my day", and express your answer as a base 10 floating-point number with 3 sig. figs (e.g. 3.45 x 10-10). Show your work! (What did you start with in binary, and how did you get that to decimal?) 3 pts ii. What's the ratio of machine precision now to machine precision then? (i.e. how much more accurately can we store numbers since 64-bit processors arrived in the early 2000s?)

Answer 1