Problem 2. It was demonstrated in class that the number of bits associated with

Question

Problem 2. It was demonstrated in class that the number of bits associated with the mantissa in Excel floating point numbers is 50. The text gives an equation relating the number of mantissa bits t to machine epsilon as -2". Develop a similar spreadsheet to successively add 1/2° to e to estimate the smallest number which can be added to e and have the result distinguishable from e. Calculate the relative error in describing e and compare it to Eni. Explain why the smaller of the two should be so. n t as e - 2 Develop a sinular

Explanation / Answer

Solution:

2.718281828

We observe that the value of e + 2-n (column 3) remains constant for all n > 35 whereas the value of epsilonm (column 2) keeps decreasing even beyond 2-40.

44.70118449

0.022370772

The last two columns show the value of e3.8 and e-3.8.

We take inverse of e3.8.

(e3.8)-1 = 0.022370772

Incidentally, using Excel the value of 1/3.8 and the value of -3.8 using power series expansion coincide

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n 2-n e + 2-n 0 1 3.718281828 -1 0.5 3.218281828 -2 0.25 2.968281828 -3 0.125 2.843281828 -4 0.0625 2.780781828 -5 0.03125 2.749531828 -6 0.015625 2.733906828 -7 0.0078125 2.726094328 -8 0.00390625 2.722188078 -9 0.00195313 2.720234953 -10 0.00097656 2.719258391 -11 0.00048828 2.71877011 -12 0.00024414 2.718525969 -13 0.00012207 2.718403899 -14 6.1035E-05 2.718342864 -15 3.0518E-05 2.718312346 -16 1.5259E-05 2.718297087 -17 7.6294E-06 2.718289458 -18 3.8147E-06 2.718285643 -19 1.9073E-06 2.718283736 -20 9.5367E-07 2.718282782 -21 4.7684E-07 2.718282305 -22 2.3842E-07 2.718282067 -23 1.1921E-07 2.718281948 -24 5.9605E-08 2.718281888 -25 2.9802E-08 2.718281858 -26 1.4901E-08 2.718281843 -27 7.4506E-09 2.718281836 -28 3.7253E-09 2.718281832 -29 1.8626E-09 2.71828183 -30 9.3132E-10 2.718281829 -31 4.6566E-10 2.718281829 -32 2.3283E-10 2.718281829 -33 1.1642E-10 2.718281829 -34 5.8208E-11 2.718281829 -35 2.9104E-11 2.718281828 -36 1.4552E-11 2.718281828 -37 7.276E-12 2.718281828 -38 3.638E-12 2.718281828 -39 1.819E-12 2.718281828 -40 9.0949E-13

2.718281828

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