Exercise 1: Reconstructing 1.0 From Its Bit Decomposition — Possible Solution
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THE VERIFIED DECOMPOSITION
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1.0 decomposes as: sign = 0, raw exponent = 1023, mantissa = 0.

STEP-BY-STEP RECONSTRUCTION
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The formula is:

    (-1)^sign * (1 + mantissa/2^52) * 2^(exponent - 1023)

Substituting the verified values:

  Step 1 - sign term:
    (-1)^0 = 1
    (a positive number, as expected for 1.0)

  Step 2 - mantissa term:
    1 + mantissa/2^52
    = 1 + 0/2^52
    = 1 + 0
    = 1
    (the mantissa is all zero bits, so the "1 + fraction" term is just
    exactly 1 - this is what an all-zero mantissa means: no correction
    beyond the implicit leading 1 bit)

  Step 3 - exponent term:
    2^(exponent_raw - 1023)
    = 2^(1023 - 1023)
    = 2^0
    = 1

  Step 4 - combine all three:
    1 * 1 * 1 = 1

RESULT
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The formula reconstructs exactly 1.0, confirming that the simplest
possible case - a mantissa of all zeros and an exponent exactly
equal to the bias - represents precisely the number 1.0, with the
implicit leading bit doing all the work and no rounding involved at
all.

WHY THIS WORKS AS AN ANSWER
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The answer walks through all three multiplicative terms of the
formula individually rather than just plugging in numbers and
asserting the result, and explicitly explains what an all-zero
mantissa means in terms of the "1 + fraction" reconstruction, tying
the arithmetic back to the chapter's own conceptual explanation of
the implicit leading bit.