Certified dyadic enclosures for unit-circle phases

Input an exact rational $q$, an exact rational target $0<\tau \le 1\mathrm{/}4$, integer dyadic exponent limits $e_{\min }\le e_{\max }$, an initial precision $p_0\ge 2$, a finite precision cap $p_{\max }\ge p_0$, and a finite elementary-series cap $J\ge 1$. The target value is $w=e^{\pi iq}=\cos (\pi q)+i\sin (\pi q)$ by $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$, for the canonical Fourier $\pi$ identified with the interval evaluator's constant by The interval evaluator's arctangent definition uses the canonical Fourier pi. For each precision $p=p_0,p_0+1,\dots ,p_{\max }$, and $t\in \{1\mathrm{/}5,1\mathrm{/}239\}$, form the exact rational interval $I_J(t)=[A_J(t)-R_J(t),A_J(t)+R_J(t)]$, where $A_J(t)={\sum }_{j=0}^J\frac{(-1{)}^j{t}^{2j+1}}{2j+1},R_J(t)=\frac{\mathrm{\mid }t{\mathrm{\mid }}^{2J+3}}{(2J+3)(1-t^2)}\mathrm{.}$ The Machin identity gives the interval $P=16I_J(1\mathrm{/}5)-4I_J(1\mathrm{/}239)$ containing $\pi$. Form the exact rational interval $qP$ by taking the minimum and maximum of its two endpoint products, then round its endpoints down and up to obtain a dyadic input interval $\mathrm{\Theta }$ in format $(p,e_{\min },e_{\max })$. If this conversion overflows, treat that precision as a non-finite evaluation. Otherwise build two finite real expression DAGs with input interval $\mathrm{\Theta }$ and requested outputs $\cos (\mathrm{\Theta })$ and $\sin (\mathrm{\Theta })$, and call Certified range evaluation for elementary real and complex functions using the same dyadic format and truncation index $J$; its fixed finite-cell rules govern range reduction. If either result is non-finite, continue at the next precision unless the precision cap is reached, in which case return inconclusive with the supplier status. If the results are finite intervals $C=[c_{-},c_{+}]$ and $S=[s_{-},s_{+}]$, set $\widehat{w}=c_{-}+i{s}_{-}$ and compute an upward-rounded dyadic $ϵ\ge (c_{+}-c_{-})+(s_{+}-s_{-})$. If $ϵ\le \tau$, return certified(\widehat w,\epsilon); otherwise continue or return inconclusive at the cap. A non-finite endpoint or overflow returns its explicit failure status. The routine makes no finite-error claim on a failure branch.

The lower-left rectangle corner is already a representable dyadic pair. The radius uses the sum of coordinate widths, which bounds complex modulus error without a square-root computation. Precision refinement is finite and may return inconclusive; only the certified branch is used by dependent algorithms.