This batch covers native math::sec, csc, cot, sech, csch, coth, and catalan<Real>. AGM and factorial are documented separately. The existing MPFR overloads and their explicit-precision contracts are unchanged. Native inputs must stay native, including configured binary80 and binary128.

Contracts and numerical acceptance

Trigonometric reciprocals evaluate at the original represented input. The rounded native value of pi/2 is not a mathematical pole, so tests must compare against that input rather than expect an artificial infinity there. Native sine/cosine/tangent supply argument reduction; this batch adds no phase approximation. Hyperbolic reciprocal tails must remain nonzero whenever the rounded result is representable, even if native sinh/cosh would overflow.

Region / all configured native formats	Measure	Intended criterion	Guarantee level	Observations
Ordinary finite arguments	Absolute error in reference-result spacing, MPFR512 oracle	<=4 units	Empirical target	See measurements below
Near trigonometric poles/zeros; very large arguments	Same exact-input reference measure; distinguish input conditioning from evaluation error	<=4 units	Empirical target, no uniform relative claim about an unrounded argument	See measurements below
Hyperbolic tails, minimum normal/subnormal results and range boundaries	Absolute error with subnormal spacing clamped; rounded value classification	<=4 units and matching sampled zero/infinity classification	Empirical target	See measurements below
Signed zeros, infinities, NaNs	Exact value/sign/classification	Mathematical reciprocal convention	API contract	See measurements below
Native Catalan constant	Compare with independently rounded wider reference	Correct rounding at each configured native format	Specific constant guarantee, not a general function guarantee	See measurements below
Controlled independent values	Error compared with the preceding precision	At least 16-fold improvement	Demonstrated examples	See measurements below

A spacing unit is 2^(max(e,min_exponent)-digits) for native results, with e the unrounded reference's binary exponent. Computation assumes nearest rounding and gradual underflow. Tests cover both sides of every algorithm join.

At signed zero, sec/sech return 1; csc/cot/csch/coth return signed infinity. At infinity the trigonometric reciprocals return NaN, sech returns positive zero, csch returns signed zero, and coth returns the sign of its argument. NaN propagates through all six functions. No errno or exception-flag promise is added. Existing MPFR exact-state rules remain in its scalar layer.

Formulas and range handling

Sec, csc and cot use 1/cos(x), 1/sin(x) and 1/tan(x). Coth uses 1/tanh(x) without an overflowing intermediate. These compositions are validated against the acceptance criteria; no universal correctly-rounded claim is made.

For sech/csch, use the native cosh/sinh reciprocal while abs(x) <= digits. The providers remain well inside their exponent range there. Beyond that join, the denominator correction in 2*exp(-abs(x))/(1 +/- exp(-2*abs(x))) is smaller than epsilon squared, so evaluate the common tail 2*exp(-abs(x)). Use that direct form while abs(x) < -min_exponent*log_two<Real>; the exponential is still normal with a conservative margin. Farther out, compute h = exp(-abs(x)/2) and h*(2*h). This preserves final subnormals even when exp(-abs(x)) itself would prematurely round to zero. Restore the sign for csch. The threshold chooses a formula, not the expected range classification.

An initial half-exponential formula with its full denominator immediately above abs(x)=1 reached 4.107 spacing units for binary80 near x=-1.00000000000000000477048955893621950963279232. The wider direct-provider region avoids amplifying that exponential rounding error. That original input and both neighbors of every selected join are regression tests.

Catalan is assembled at compile time from a sufficiently long binary prefix expressed as exact integer chunks and powers of two. Tests verify the final native rounding against a wider oracle and use the alternating series sum (-1)^k/(2k+1)^2, accelerated by Euler's transformation, as an independent test reference. MPFR's constant routine generates/checks the prefix offline; it is not a native runtime dependency.

The prefix is 0xea7cb89f409ae845215822e37d32d0c6 * 2^-128, split into 64-bit integer chunks. Multiplication by binary scale factors and the final native sum avoid a double intermediate. At 1024-bit reference precision, 400 terms of the Euler-transformed alternating series give an independent check; the positive transformed coefficients are at most one, so its omitted tail is bounded by 2^-400. The final constants match the correctly rounded wider values for all four native formats.

Observations and validation

GCC 13/glibc 2.39, 512-bit MPFR oracle: roughly 39,000–41,000 deterministic inputs per format, including ordinary grids, tiny inputs, pole/join neighbors, full exponent ranges and fixed-seed random significands. Observed maxima in spacing units, rounded upward:

Format	sec	csc	cot	sech	csch	coth
float32	1.431	1.446	1.453	2.072	2.206	2.687
float64	1.394	1.407	1.428	2.323	2.080	1.889
binary80	2.038	1.856	1.580	2.560	2.560	1.702
binary128	2.165	2.013	1.522	2.016	1.953	1.935

No sampled range-classification failures remained. These are measurements, not a replacement for the intended four-unit criterion or a global proof. The integrated probes record their own maxima and worst inputs in test XML. Independent sine/cosine/sinh/cosh series check precision improvement at 1/4; identities and core boundary cases also run with MPFR disabled.

Targeted tests passed: 70 tests including reciprocal functions, constants, existing scalar/special-function cases and an oracle regression. The oracle's binary-value reconstruction now preserves negative zero and nonfinite values; previously it discarded the sign of zero. A separate MPFR-disabled Release check passed all 12 native reciprocal/constant tests, and the example ran in that configuration.

Full local validation passed with examples, CLI and Python enabled:

Configuration	Passed	Skipped
GCC 13 Debug, MPFR/MPC and MPLAPACK, native binary128	3182	1
GCC 13 Release, MPFR disabled, native binary128	2427	1
Clang 19 Debug, MPFR/MPC and MPLAPACK, binary128 disabled	2863	1

The skip in each configuration was the single-node NUMA topology test. Unfiltered numerical suites and their coverage reports also passed. Clang's cell is not binary128 evidence; macOS and CUDA execution were not exercised.

Run native_reciprocal_math_example to see the retained hyperbolic tails and native Catalan constants. No external dependency or provider selection changes are introduced by this batch.