Exercise 3: Why Wrapping an Accumulating Angle Is a Real Accuracy Improvement — Possible Solution
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WHY LARGE ANGLE VALUES HAVE LESS USABLE PRECISION
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A standard double-precision float allocates a fixed number of bits
(52, per Numerical Methods & Floating-Point Computation Chapter 2) to
represent a number's significant digits, regardless of that number's
overall magnitude - but the ABSOLUTE SIZE of the smallest
representable gap between two adjacent values grows as the number
itself grows larger, since the same number of significant bits has to
cover a wider range. A value near 27,400 therefore has a much coarser
spacing between representable values than a value near 2*pi (about
6.28) - every unit of precision is "spread thinner" at the larger
magnitude.

WHY THIS DIRECTLY EXPLAINS THE VERIFIED RESULT
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This chapter's own experiment accumulated the exact same sequence of
small increments two different ways: once letting the angle grow
unboundedly to about 27,400 radians, and once wrapping it back into
the range [0, 2*pi) after every single addition. Every individual
addition in both cases introduces some small rounding error, but in
the unwrapped version, that rounding error is being added to an
increasingly large number with increasingly coarse precision, so each
successive addition loses more absolute precision than the last. In
the wrapped version, every addition is performed on a number that
stays confined to a small, consistent range near 2*pi, so each
addition's rounding error stays small and consistent throughout the
entire 2,000,000-frame simulation, rather than growing. This directly
produced the verified nearly-four-orders-of-magnitude accuracy
difference (about 3.15*10^-7 relative error unwrapped vs. about
4.54*10^-11 wrapped).

WHY THIS ISN'T JUST ABOUT CODE STYLE
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It might seem like wrapping an angle is only a matter of keeping
numbers "tidy" or avoiding overflow - but this chapter's own verified
numbers show it has a real, measurable effect on the actual accuracy
of the resulting sin() and cos() values used to render or simulate
the object's orientation. A rotating object whose angle is never
wrapped doesn't just have an inconveniently large angle variable - it
genuinely renders or simulates less accurately over time, purely as a
side effect of floating-point representation, even though the
underlying rotation logic (adding a fixed increment each frame) never
changed.

WHY THIS WORKS AS AN ANSWER
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The explanation connects the specific mechanism (a double's precision
being relative to its own magnitude, from the prerequisite course) to
this chapter's own concrete verified numbers, and explicitly argues
against treating the wrapping practice as merely stylistic, since the
chapter's own experiment demonstrates a real, quantifiable accuracy
cost to skipping it.