Range controls, roots, and reusable-denominator division
Public exponent range
Each execution context stores:
\[
\operatorname{emin}\le e\le\operatorname{emax}.
\]
The short-form range and flag functions operate on the calling thread's
default context. Programs using explicit contexts can continue using the
ppf_context_* interfaces.
ppf_check_range assumes its input is the correctly rounded result in an
extended exponent range together with the corresponding ternary direction. It
then applies the current range, updates the value when overflow or underflow
occurs, propagates the ternary relation to the exact mathematical result, and
raises the required flags.
ppf_subnormalize emulates a fixed-point/subnormal system without adding a
subnormal value category to ppf_t. It rounds to the effective precision
implied by the current exponent and emin, using the incoming ternary value to
avoid double-rounding errors.
When this step reduces the effective precision, it raises PPF_FLAG_UNDERFLOW
even if the value remains exact, matching mpfr_subnormalize.
PPF_FLAG_INEXACT is raised only when the returned ternary value is nonzero.
This flag rule is specific to subnormal emulation; ordinary arithmetic uses
the context's exponent range and its usual rounding rules.
General roots
For a positive exact dyadic input
\[
x=M2^e,
\]
and degree \(n\), the implementation chooses a scale \(S\), constructs an integer radicand, and calculates
\[
a=\left\lfloor R^{1/n}\right\rfloor
\]
with an exact remainder. A nonzero remainder proves
\[
a2^{-S}<x^{1/n}<(a+1)2^{-S}.
\]
The two endpoints are passed to the common interval-rounding engine. If the interval does not yet decide the destination, the scale is increased and the calculation is repeated.
Odd roots of negative values negate the positive interval. Even roots of
negative values produce NaN. Signed zero, infinity, reciprocal degrees, and
the older root zero-sign convention are handled separately.
ppf_hypot forms the exact value
\[
x^2+y^2
\]
and rounds its square root once. It does not round two squares and an intermediate sum independently.
Reusable denominator plans
A plan owns an exact normalized copy of the denominator and the destination precision. For reusable workloads it also stores a portable reciprocal estimate.
If \(D\) is the normalized positive denominator, the plan constructs
\[
R=\left\lfloor\frac{2^S}{D}\right\rfloor.
\]
For a suitably scaled numerator \(N\), it estimates
\[
q_0=\left\lfloor\frac{NR}{2^S}\right\rfloor.
\]
The estimate is exact or one below the true quotient. The implementation forms
\[
r_0=N-q_0D.
\]
When \(r_0\ge D\), one exact correction produces
\[
q=q_0+1,\qquad r=r_0-D.
\]
Otherwise \(q=q_0\) and \(r=r_0\). The exact remainder supplies sticky information for the final floating-point rounding decision.
The plan is immutable after construction. Independent numerators may use it concurrently. Each worker receives its own local context and scratch data, and flags are merged into the caller's context after the batch.
Aliasing and failure behavior
A planned output may alias its corresponding numerator. Cross-index aliasing in a parallel batch is rejected because concurrent writes would make the result order-dependent.
No calculation in this subsystem delegates to MPFR. Unsupported resource or
representation cases return PPF_TERNARY_ERROR and raise the appropriate
context status rather than silently switching libraries.