Question

(30 pts) In addition to the default IEEE double-precision format (8 byte 64 bits) to store floating-point numbers, MATLAB can also store the numbers in single-precision format (4 bytes, 32 bits). Each value is stored in 4 bytes with 1 bit for the sign, 23 bits for the mantissa, and 8 bits for the signed exponent: Sign Signed exponent Mantissa 23 bits L bit 8 bits Determine the smallest positive value (expressed in base-10 number) that can be represented using the single-precision IEEE normalized floating-point number format. What is its machine epsilon under this format? (Show your work)

Answer 1

Smallest positive:
--------------------
0 00000001 00000000000000000000000
sign bit is 0(+ve)
exp bits are 00000001 => in decimal it is 1
so, exponent/bias is 1-127 = -126
frac bits are

Decimal value is 1. * 2^-126
1. in decimal is 1
1. * 2^-126 in decimal is 1.1754943508222875e-38
so, 00000000100000000000000000000000 in IEEE-754 single precision format is 1.1754943508222875e-38
smallest positive value is 2^-126 which is 1.1754943508222875e-38

machine epsilon is 2^-23 because mantissa is of length 23