Why the arithmetic is outward

The lower binomial tail in (eq:cpL) is increasing in its probability argument. The lower side of its root bracket can only decrease the lower confidence endpoint. The CDF in (eq:cpU) is decreasing; the upper side can only increase the upper endpoint. The square-root routine returns an upper bracket.

After $32$ bisections, each root bracket has width $2^{-32}$. For the three-setting lower fidelity, outward endpoint loss is at most $3\cdot 2^{-33}$ before clipping. After $40$ square-root bisections, the untruncated doubled square-root upper expression has an extra numerical allowance of at most $2^{-39}$. These are deterministic arithmetic bounds, distinct from statistical confidence.

The binomial recurrence's intermediate terms are integers

when $p=a/b$. The asserted exact division advances this term to the next one. The optional direct-sum test checks this identity over a finite set; the identity itself holds algebraically.

The exact function tests the mathematical expression using the conservative returned $F_L$. It need not coincide at an extremely close boundary with testing the separately rounded-up display . Both conventions are conservative if their numerical meaning is declared. For a laboratory implementation, archive which rule is used.