[rmlarsen]

Summary

Stacked on #12641 and on #12568 below it. For six fp32mp2 function families this adds a second, shorter kernel and selects between the two by accuracy level, the way #12641 already does for fmod:

Family	fp32mp2_low and fp32mp2 (default)	fp32mp2_high
erfc	#12641's series below 1/2, one reciprocal and a degree-18 polynomial above	#12641's kernel
erfcx, normcdf	the one-reciprocal kernel of erfc	#12641's kernel
sin, cos, tan, sincos, sinpi, cospi, sincospi	one-fma fast path, minimax kernels	#12641's kernels
asin, acos, atan, atan2	one table-driven kernel	#12641's polynomial kernels
exp, exp2, exp10, log, log2, log10	on the tables of pow, at every level	same
expm1, log1p	on the tables of pow	the same, plus a Taylor series for |x| < 2⁻⁷
sinh, tanh, atanh	on the exp and log kernels	#12641's formulas on those kernels
cosh, asinh, acosh	on the exp and log kernels, at every level	same
sqrt, rsqrt	#12641's kernels	a second-order kernel
norm3d, norm4d, rnorm3d, rnorm4d	a pair kernel, no FP64	#12641's double kernels
rhypot	the pair kernel, at every level	same

The default level gets the faster kernel and high the more accurate one; for sqrt and rsqrt that means #12641's kernel stays the default. Where the high kernel is #12641's, its results are #12641's bit for bit on every range tested; erfcinv and normcdfinv are not in the table but call log, and change with it at every level (see Accuracy).

One commit per family, each with its own lit test at all three levels (erfcx and normcdf have a commit and a test of their own).

The kernels added here carry the edge-case fixes of #12568 where they apply: erfcx at every level is finite up to the argument where 2·exp(x²) overflows (x = −0x1.2c3cbcp+3, decided by its low limb) and returns a subnormal number instead of 0 for x ≥ 2¹²⁷ on the device, and the default-level sin returns a zero argument with its sign only when both limbs are zero. The fp64mp2 sinh, tanh, asinh, atanh, expm1 and log1p return a zero argument unchanged only when both limbs are zero, as #12568 does for atan, asin and atan2. atan2 reads an operand with a zero high limb over a nonzero low limb as its low limb, as in #12568, at every level: the entry code is shared by the two kernels.

The fp32mp2 kernels of expm1, log1p, sinh, tanh, asinh and atanh take the size of their argument, and with it the result f(x) = x of a tiny one, from the high limb. At the low level an argument whose low limb is above 2⁻²⁰ of its high limb is therefore renormalized before the kernel: a low-accuracy pair with a zero high limb over a large low limb, such as (2²⁴, 0.25) − 2²⁴ = (0, 0.25), gets the result of 0.25 (expm1: 0.284), as in #12568.

Throughput

Speedup over #12641 at 8989bc7 with the contraction fix (the code of its head, c330b4c) and the cbrt seed of #12568, that is, before that PR's special-value fixes; their own cost is listed there and is included in the high rows below. Device: evaluations per second of a kernel that calls the function 2048 times per thread on 2²⁰ arguments, best of 7 launches. Host: ns per call on one core, 2048 arguments, fastest of 40 rounds (M4: 300 rounds, minimum of six alternating passes). Every measurement ran alone on an otherwise idle machine, base and branch alternating, in two passes that agree within 3% unless a range is given.

• A: NVIDIA RTX PRO 6000 Blackwell (sm_120) and AMD Ryzen 7 7800X3D, CUDA 13.3, GCC 15.2 -O3 -march=x86-64-v3
• B: NVIDIA RTX 6000 Ada Generation (sm_89) and AMD Ryzen Threadripper PRO 7965WX, CUDA 13.4, GCC 15.2, same flags
• M4: Apple M4, Apple clang 21 -O3 -mcpu=apple-m4

Default level

Function, range	GPU A	GPU B	x86 A	x86 B	M4
erfc [0, 1/8]	0.99×	1.00×	0.99×	0.90–0.99×	1.01×
erfc [-4, 4]	1.20×	1.12–1.25×	1.68×	1.68×	1.79×
erfc [4, 8.2]	1.28×	1.25–1.36×	1.72×	1.72×	1.83×
sin / cos / tan [-0.78, 0.78]	1.62 / 1.60 / 1.26×	1.54 / 1.57 / 1.20×	1.47 / 1.58 / 1.52×	1.49 / 1.51–1.59 / 1.39×	1.28 / 2.36 / 1.13×
sin / cos / tan [-100, 100]	1.52 / 1.50 / 1.26×	1.45 / 1.46 / 1.23×	1.77 / 1.73 / 1.52×	1.61–1.79 / 1.76 / 1.53×	1.49 / 2.22 / 1.38×
sin [2·10⁶, 10³⁰]	1.12×	1.23×	1.33×	1.34×	1.09×
sinpi / cospi [-2, 2]	1.13 / 1.12×	1.12 / 1.13×	1.16 / 1.17×	1.17 / 1.18×	1.15 / 1.17×
asin / acos [-1, 1]	3.43 / 3.74×	3.18–3.45 / 3.26×	2.14 / 2.41×	2.15 / 2.16×	1.89 / 1.92×
atan [-1, 1] / [1, 1000]	2.15 / 2.14×	2.19 / 2.20×	2.83 / 2.93×	2.58 / 2.62–2.92×	3.35 / 3.19×
atan2 [-10, 10]	2.24×	2.24×	2.86×	2.88×	2.45×
exp / exp2 / exp10 [-10, 10]	2.29 / 2.28 / 2.04×	1.73 / 1.84 / 1.66×	3.06 / 3.80 / 2.96×	3.11 / 3.49 / 2.70×	2.40 / 2.77 / 2.32×
expm1 [-1, 1] / [-10, 10]	2.95 / 2.86×	2.40 / 2.20×	1.89 / 2.05×	1.72 / 2.08×	1.75 / 1.85×
log / log2 / log10 [10⁻¹⁰, 10¹⁰]	1.28 / 1.24 / 1.19×	1.29 / 1.29 / 1.26×	2.97 / 2.48 / 2.32×	2.73 / 2.25 / 2.11×	2.66 / 2.57 / 2.41×
log1p [-0.5, 1] / [10⁻¹⁰, 10¹⁰]	1.42 / 1.42×	1.67 / 1.73×	2.38 / 2.07×	2.16 / 1.90–2.11×	2.30 / 2.00×
sinh [-1, 1] / [-20, 20]	0.87 / 2.28×	0.88 / 1.76×	1.48 / 2.28×	1.51 / 2.11–2.35×	1.18 / 1.83×
cosh [-20, 20], tanh [-5, 5]	2.14, 1.71×	1.60, 1.68×	2.42, 3.08×	2.22, 2.82–3.13×	2.29, 2.51×
asinh / acosh / atanh	1.29 / 1.21 / 1.35×	1.37 / 1.33 / 1.31×	2.33 / 1.73 / 1.99×	2.34 / 1.74 / 1.86–2.01×	2.78 / 2.0 / 2.07×
sqrt, rsqrt, hypot	1.00–1.02×	0.99–1.00×	0.97–0.99×	1.00×	1.00×
rhypot	1.41×	1.05×	1.20×	1.20×	1.12×
norm3d / rnorm3d / norm4d / rnorm4d	43 / 41 / 51 / 48×	19 / 19 / 19 / 19×	1.32 / 1.41 / 1.35 / 1.42×	1.30 / 1.24–1.38 / 1.21–1.33 / 1.25×	not measured

fp32mp2_high

Function	GPU A	GPU B	x86 A	x86 B	M4
erfc, trig, sinpi, cospi	0.96–1.05×	0.93–1.01×	0.98–1.09×	0.88–1.15×	0.99–1.01×
asin, acos, atan	0.95–1.00×	0.93–1.01×	0.96–1.00×	0.97–1.00×	0.99–1.02×
atan2	0.91×	0.94×	1.02×	1.02×	0.90×
exp, exp2, exp10, log, log2, log10	as at the default level
expm1 [-1, 1] / [-10, 10]	2.28 / 2.48×	2.00 / 2.10×	1.89 / 2.04×	1.72 / 1.87×	1.76 / 1.85×
log1p [-0.5, 1] / [10⁻¹⁰, 10¹⁰]	1.32 / 1.21×	1.44 / 1.30×	2.36 / 2.27×	2.16–2.41 / 2.08×	2.29 / 2.14×
sinh [-1, 1] / [-20, 20]	1.00 / 1.42×	1.03 / 1.29×	0.98 / 1.98×	0.99 / 1.78×	1.01 / 1.77×
tanh, atanh	1.25, 1.27×	1.26–1.32, 1.27×	1.81, 1.48×	1.82, 1.49×	1.60, 1.64×
sqrt / rsqrt	0.43 / 0.52×	0.48 / 0.58×	1.03 / 0.93×	1.03 / 0.93×	0.92 / 0.94×
norms	1.00×	1.00×	0.78–0.86×	0.75–0.93×	not measured

erfcx and normcdf were measured separately and differently. Host: ns per call on an 11th Gen Intel Core i5-11500T, GCC 15.2 -O3 -march=x86-64-v3, one core of an otherwise idle machine, median of three alternating passes. Device: executed instructions per call (ncu, smsp__inst_executed) of a kernel on sm_89 whose 2²⁰ arguments mix the cases of the function, not time.

Function, range	x86, default	x86, high	Device instructions, default	Device instructions, high
erfcx [-3, 10]	102.9 → 67.0 ns (1.54×)	103.0 → 106.8 ns (0.96×)	35.2 → 19.6 (1.80× fewer)	35.2 → 21.6 (1.63× fewer)
normcdf [-5, 5]	151.2 → 109.7 ns (1.38×)	1.00×	22.5 → 20.4 (1.10× fewer)	unchanged

The device counts are with #12568's erfcx fixes (0.2 to 0.3 instructions per call more); the x86 times were taken before them. erfcx executes fewer instructions at high too, with the same results: its exp and its kernel are now each written once and shared by the cases that need them, where a warp whose threads take different cases used to execute a copy per case.

Slower than #12641:

• sinh on [-1, 1] at the default level, on the GPU (0.87×). The shorter kernel is one path for all arguments; [fp] fp32mp2 math: native float-float implementations to 45-48 bits, exact pair rounding and remainder, C99 special values #12641's Taylor series is cheaper for small ones. It is faster on every host and on wider ranges.
• sqrt and rsqrt at high on the GPU (0.43× and 0.52×). That is the price of the second-order step (2.4 → 0.5 u² and 7.7 → 2.6 u²). The default level keeps [fp] fp32mp2 math: native float-float implementations to 45-48 bits, exact pair rounding and remainder, C99 special values #12641's kernels and timings.
• erfcx at high on x86 (0.96×). The price of the shared exp on the host; on the device it executes 1.63× fewer instructions.
• atan, atan2 and the norms at high. These are [fp] fp32mp2 math: native float-float implementations to 45-48 bits, exact pair rounding and remainder, C99 special values #12641's kernels with the entry code of [fpmp] fp32mp2 math on top of #12641: contraction-independent host results, special-value fixes, faster host cbrt seed #12568, and the cost is that PR's, not this one's.

Accuracy

Maximum relative error against MPFR in u² (u = 2⁻²⁴), 2M random pairs with full low limbs per range, on the M4. For the three erfc rows the entry is maximum / mean.

Function, range	#12641	default	high
erfc [0, 1/8]	0.88 / 0.10	0.88 / 0.10	0.88 / 0.10
erfc [-4, 4]	4.16 / 0.28	9.53 / 0.47	4.16 / 0.28
erfc [4, 8.2]	4.40 / 0.46	10.23 / 1.06	4.40 / 0.46
sin [-0.78, 0.78] / [-100, 100] / [2·10⁶, 10³⁰]	1.89 / 3.16 / 3.84	1.92 / 3.61 / 5.49	1.89 / 3.16 / 3.84
cos [-0.78, 0.78] / [-100, 100]	3.26 / 3.22	4.23 / 4.55	3.26 / 3.22
tan [-0.78, 0.78] / [-100, 100]	5.36 / 6.01	5.99 / 6.51	5.36 / 6.01
sinpi / cospi [-2, 2]	3.23 / 3.18	4.42 / 3.92	3.23 / 3.18
asin / acos [-1, 1]	2.38 / 3.33	5.71 / 5.72	2.38 / 3.33
atan [-1, 1] / [1, 1000]	1.80 / 1.71	3.85 / 3.92	1.80 / 1.71
atan2 [-10, 10]	3.18	5.40	3.18
exp / exp2 / exp10 [-10, 10]	3.23 / 2.55 / 2.96	2.85 / 2.72 / 2.93	same as default
expm1 [-1, 1] / [-10, 10]	2.76 / 3.24	3.91 / 3.51	3.91 / 3.51
expm1 [-2⁻⁷, 2⁻⁷]	1.88	4.04	1.88
log [0.5, 2] / [10⁻¹⁰, 10¹⁰]	6.76 / 5.30	4.31 / 4.70	same as default
log2 / log10 [10⁻¹⁰, 10¹⁰]	5.73 / 7.43	3.94 / 4.05	same as default
log1p [-0.5, 1] / [10⁻¹⁰, 10¹⁰]	6.28 / 5.34	5.70 / 6.31	5.70 / 5.56
log1p [-2⁻⁷, 2⁻⁷] / [2⁻⁷, 2⁻⁴]	1.79 / 2.06	8.86 / 7.76	1.79 / 7.76
sinh [-1, 1] / [-20, 20]	2.69 / 3.39	4.42 / 5.31	2.69 / 3.62
cosh [-20, 20]	3.11	3.38	3.38
tanh [-5, 5]	2.21	6.53	2.21
asinh [-100, 100] / acosh [1, 100]	4.83 / 5.02	4.91 / 4.63	same as default
atanh [-1, 1]	2.74	7.93	5.30
sqrt / rsqrt [10⁻¹⁰, 10¹⁰]	2.41 / 7.71	2.41 / 7.71	0.50 / 2.59
hypot / rhypot [10⁻⁵, 10⁵]	2.83 / 7.71	2.83 / 7.71	2.83 / 7.71
norm3d / norm4d [10⁻⁵, 10⁵]	0.55 / 0.58	5.04 / 4.33	0.55 / 0.58
rnorm3d / rnorm4d [10⁻⁵, 10⁵]	0.55 / 0.56	7.65 / 8.09	0.55 / 0.56

Where the default level is less accurate than #12641, high restores #12641's error, except:

• log1p on [2⁻⁷, 2⁻⁴] is 7.8 u² at both levels ([fp] fp32mp2 math: native float-float implementations to 45-48 bits, exact pair rounding and remainder, C99 special values #12641: 2.1 u²).
• atanh is 5.3 u² at high ([fp] fp32mp2 math: native float-float implementations to 45-48 bits, exact pair rounding and remainder, C99 special values #12641: 2.7 u²).
• expm1 on [-1, 1] is 3.9 u² at both levels ([fp] fp32mp2 math: native float-float implementations to 45-48 bits, exact pair rounding and remainder, C99 special values #12641: 2.8 u²).
• The reciprocal norms reach 13 u² on the device at the default level, whose reciprocal square root seed is approximate.
• erfcinv for arguments down to 2⁻¹⁰⁰ is 2.1 u² at every level ([fp] fp32mp2 math: native float-float implementations to 45-48 bits, exact pair rounding and remainder, C99 special values #12641: 1.3 u²): it calls log. normcdfinv calls it too, keeps its error and is 1.31× faster on x86.

erfcx and normcdf, 1M pairs per range on x86-64: erfcx is 7.3 u² on [-3, 10] and 5.3 u² on [8, 2⁶⁰] at the default level (#12641 and high: 3.6 and 3.9 u²); normcdf is 6.7 u² on [-13, 0] and 7.0 u² on [-1, 1] (#12641 and high: 4.5 and 3.8 u²). On the device the largest errors are 6.1 u² (erfcx) and 7.6 u² (normcdf). The same run has 11.5 u² for erfc at x ≈ 5.55, above the 10.23 of the table's 2M pairs on [4, 8.2].

At the default level the host and an sm_89 GPU give different last bits for part of the arguments of asin and acos (1–14% per range), erfc, erfcx and normcdf (up to 10%), atanh (up to 5%) and tanh (up to 2%): their kernels start from the device's approximate reciprocal and reciprocal square root. The largest error is the same on both. #12641 has the same for rsqrt, cosh, asinh and boys_f0.

exp, log, log2 and log10 are at least as accurate as #12641 in maximum error, at every level.

Instructions

fp32 SASS instructions of a kernel that calls the function once, #12641 → default / high, CUDA 13.4:

Function	sm_75	sm_89	sm_120
erfc	680 → 576 / 680	688 → 584 / 688	688 → 584 / 688
sin, cos	728 → 664 / 728	720 → 656 / 728	760 → 696 / 760
tan	736 → 704 / 744	736 → 696 / 744	776 → 736 / 776
sinpi / cospi	368 → 344 / 368, 312 → 288 / 312	368 → 344 / 368, 312 → 288 / 312	376 → 352 / 376, 328 → 304 / 328
asin / acos	456 → 144 / 472	464 → 152 / 480	464 → 152 / 480, 480 → 152 / 496
atan / atan2	248 → 128 / 264, 320 → 160 / 320	256 → 136 / 272, 328 → 168 / 328	264 → 136 / 272, 328 → 168 / 336
exp / exp2 / exp10	152 → 96, 136 → 88, 152 → 104	160 → 104, 144 → 96, 160 → 112	160 → 112, 144 → 96, 160 → 112
expm1 / log1p	288 → 208 / 256, 288 → 296 / 352	288 → 216 / 264, 296 → 304 / 360	288 → 216 / 272, 296 → 288 / 344
log / log2 / log10	152 → 128, 152 → 128, 160 → 136	160 → 136, 160 → 136, 168 → 144	168 → 144, 168 → 152, 168 → 160
sinh / cosh / tanh	304 → 160 / 240, 192 → 120, 336 → 240 / 288	312 → 168 / 248, 200 → 128, 344 → 248 / 296	304 → 176 / 256, 192 → 128, 344 → 248 / 296
asinh / acosh / atanh	512 → 552, 504 → 568, 472 → 544 / 800	520 → 552, 512 → 576, 480 → 544 / 816	528 → 552, 520 → 568, 488 → 536 / 784
sqrt / rsqrt	32 → 32 / 88, 32 → 32 / 80	40 → 40 / 96, 40 → 40 / 88	48 → 48 / 96, 48 → 48 / 88

#12641's counts are the same at both levels. asinh, acosh, atanh and the default log1p have more instructions than #12641 and are still faster.

Testing

• New lit tests, host and device, each at fp32mp2, fp32mp2_low and fp32mp2_high with a bound that fits the level: math_erfc, math_erfcx (erfcx and normcdf), math_trig, math_invtrig, math_explog, math_hyperbolic, math_sqrt.
• Host. All fpmp/math* tests, accuracy and scalar_ops pass on the M4 at -O2, -O0 and -O2 -ffp-contract=off, at every commit of the branch at -O2, up to the sqrt commit; the erfcx and normcdf commit and the newer commits of [fpmp] fp32mp2 math on top of #12641: contraction-independent host results, special-value fixes, faster host cbrt seed #12568 were run on x86-64 only (lit, below).
• lit on the GPU. The full cuda/fp/units suite passes on sm_89: 68 passed, 2 unsupported (fpemu/float64, fpmp/atomic_dd), CUDA 13.4 and GCC 15.2.
• --use_fast_math (the CCCL_ENABLE_FASTMATH job of the PR matrix). The suite passes on sm_89 with it as well, 68 passed and 2 unsupported, and the fpmp/math* tests pass with it on sm_75 (GTX 1650 and RTX 2070). Fast math flushes subnormals to zero on the device, so every table row with a subnormal operand, or with a result below 2⁻⁷⁸, whose low limb can be subnormal, is left to the IEEE build.
• Compile modes. Header mode and _CCCL_FPMP_USE_LIB are clean at C++17 and C++20 for fp32mp2, low, high, fp64mp2 and fp64mp2_high.

Not run: MSVC, sm_75 and sm_120 lit, data-center GPUs. fp64mp2 is covered by the unit tests only; its high entry points forward to the single fp64mp2 kernel, except sqrt, rsqrt, which get the second-order kernel, and the norms, which keep the pair kernel (a double fallback would halve their precision).

Open question for #12641

fp32mp2_high selecting a different math kernel is #12641's own mechanism for fmod, remainder and remquo. This PR uses it for 22 more functions. If the levels are meant to govern arithmetic only, the alternative is to keep one kernel per function, and then each family needs a choice between the two columns above.

Merge order

#12641, then #12568, then this. It carries the stack's CI override, which is reset before merging.

Checklist

• New or existing tests cover these changes.
• The documentation is up to date with these changes.

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Summary

Summary by CodeRabbit

• New Features
  • Added high-accuracy options for more multiprecision math functions, including roots, exponentials, logarithms, trigonometric and hyperbolic functions, probability functions, norms, and remainder operations.
• Bug Fixes
  • Improved handling of signed zero, infinities, NaNs, subnormals, and values near domain and range boundaries.
  • Corrected rounding, classification, and special-value results for edge cases.
• Documentation
  • Expanded accuracy guidance and documented special-value behavior across accuracy levels.
• Tests
  • Added host and device coverage for multiprecision math accuracy and special values.

FPMP math implementations gain revised kernels and high-accuracy dispatch across several function families. The change adds pair-aware special-value and rounding behavior, expands host/device accuracy and boundary tests, updates FPMP documentation, and adds targeted CI configurations.

Changes: FPMP math implementation

Layer / File(s)	Summary
Square-root kernels and shared numerical helpers libcudacxx/include/cuda/__fp/fpmp_impl.h, libcudacxx/include/cuda/__fp/fpmp_impl_divsqrt.h, libcudacxx/include/cuda/__fp/fpmp_math_impl.h, libcudacxx/include/cuda/__fp/fpmp.h, docs/libcudacxx/extended_api/fp/fpmp_error_bounds.rst, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_sqrt.pass.cpp	Adds high-accuracy sqrt and rsqrt kernels and dispatch. Adds refined pair division and a fast mixed-Horner strategy. Adds reference-accuracy and special-value tests, and updates square-root error-bound documentation.
Exponential and logarithmic kernels libcudacxx/include/cuda/__fp/fpmp_math_impl_exp.h, libcudacxx/include/cuda/__fp/fpmp_math.h, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_explog.pass.cpp	Replaces fp32mp2 exponential and logarithmic implementations with table-driven kernels. Adds high-accuracy small-input paths and tests for precision, ranges, and special values.
Trigonometric and inverse-trigonometric kernels libcudacxx/include/cuda/__fp/fpmp_math_impl_trig.h, libcudacxx/include/cuda/__fp/fpmp_math.h, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_trig.pass.cpp, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_invtrig.pass.cpp	Expands Payne–Hanek reduction to 96 bits and adds separate minimax and high-accuracy kernels. Updates inverse-trigonometric paths and tests for accuracy and special values.
Hyperbolic kernels libcudacxx/include/cuda/__fp/fpmp_math_impl_hyperbolic.h, libcudacxx/include/cuda/__fp/fpmp_math.h, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_hyperbolic.pass.cpp	Revises hyperbolic and inverse-hyperbolic kernels, including selected high-accuracy paths and limb-aware domain handling. Adds accuracy and boundary tests.
Power, roots, and norms libcudacxx/include/cuda/__fp/fpmp_math_impl_pow.h, libcudacxx/include/cuda/__fp/fpmp_math.h, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_pow.pass.cpp, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_cbrt.pass.cpp, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_sqrt.pass.cpp	Revises pow, cbrt, rcbrt, and scaled norm implementations. Adds tests for reference accuracy, exact results, and exceptional inputs.
Error and probability functions libcudacxx/include/cuda/__fp/fpmp_math_impl_special.h, libcudacxx/include/cuda/__fp/fpmp_math.h, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_erfc.pass.cpp, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_erfcx.pass.cpp	Revises error and probability kernels, including endpoint and tail handling. Adds tests for error bounds, subnormal results, saturation, and probability endpoints.
Remainder, rounding, and pair classification libcudacxx/include/cuda/__fp/fpmp_math_impl_nearint.h, libcudacxx/include/cuda/__fp/fpmp_math_impl_classify.h, libcudacxx/include/cuda/__fp/fpmp_math_impl_manip.h, libcudacxx/include/cuda/__fp/fpmp_math_impl_lib.h, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math.pass.cpp, libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_special_values.pass.cpp	Adds exact sparse-pair remainder paths and updates rounding, manipulation, and classification behavior. Adds tests for special values, sparse pairs, and limb-aware results.
FPMP docs and CI coverage docs/libcudacxx/extended_api/fp/fpmp.rst, docs/libcudacxx/extended_api/fp/fpmp_spec.rst, ci/matrix.yaml	Updates operation and special-value documentation. Adds targeted GPU and CPU-precompile configurations for FPMP tests.

Change: Refactor

Merge Risk: Moderate · previous review 62496

The CI override must be reset before merging, or the required checks will not pass. A few pair edge cases, such as NaN low limbs and opposite-infinity pairs, behave inconsistently with the new classification predicates. The author has accepted this as a design choice. The fast-math test failure in the erfcx test has been fixed.

Security architecture risk: Minimal · up to 62496

The inspected changes alter numerical evaluation and accuracy selection without establishing a new security boundary, elevated authority, or material failure-containment risk. No introduced or worsened security-relevant architecture concern was identified.

Retained concerns
No architecture-level concerns identified.

Security review details

Security Blast Radius

• inferred — The demonstrated exposure is to callers receiving changed numerical results from existing FPMP interfaces. No inspected path connects those results to a privileged sink; security-sensitive downstream use remains outside the inspected coverage.

- [ ] Autofix · Keep fixing CodeRabbit findings and required CI, and resolving merge conflicts

Recent review info

Run configuration

• Source mode: Auto
• Repository: NVIDIA/cccl
• Branch: main
• File: .coderabbit.yaml
• Commit: 8d33eb1
• Layers: Repository YAML
• Review profile: CHILL
• Plan: Enterprise
• Run ID: 2546f780-81ec-46a9-bbc9-277a2745e0aa

Commits

Reviewing files that changed from the base of the PR and between 5a9b39b and 62496ed.

Files selected for processing: 8

• libcudacxx/include/cuda/__fp/fpmp_math_impl_classify.h
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_erfc.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_erfcx.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_explog.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_hyperbolic.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_invtrig.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_special_values.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_sqrt.pass.cpp

Included review availability: This review used your included allowance. Your plan provides up to 12 included reviews per hour; 8 remain after this review.

[coderabbitai]

Actionable comments posted: 16

Nitpick comments: 3

libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_cbrt.pass.cpp: 1

147-148: Trivial · Maintainability & Code Quality · Quick win

suggestion: Tighten the 16u^2 bound so the test detects a loss of the new accuracy.

The new kernels document 1.5u^2 (cbrt) and 1.4u^2 (rcbrt) worst case against binary128. The old device Halley step is documented at 7u^2. A regression to the old error, or worse, still passes <= 16.0. This test is the only accuracy guard for the rewritten Halley step. Use a bound of about 4.0, then check it on host and device. Keep the margin for the seed-independent claim.

libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_pow.pass.cpp: 1

114-177: Trivial · Maintainability & Code Quality · Quick win

suggestion: Add main-path cases for a = +1 and a = -1 with finite exponents.

The kernel's special-value block is skipped for a finite positive a and finite nonzero b. This includes a == 1, and a == -1 with an integer b. Both go through __internal_fpmp2_pow_log, __internal_fpmp2_pow_exp and __internal_fpmp2_exp_scale. C99 requires exactly 1 for pow(+1, b) and -1 or +1 for pow(-1, b) by parity. No test covers pow(+1, 2.5) or pow(-1, 3). Add cases such as {1, 0, 2.5, 0, 1}, {1, 0, -1e10, 0, 1} and {-1, 0, 3, 0, -1}. check_exact then also checks that lo is zero.

libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_sqrt.pass.cpp: 1

116-117: Trivial · Maintainability & Code Quality · Quick win

suggestion: Tighten the tolerance_high entries so the high level is distinguishable from the short kernel.

The header comment documents the high-level sqrt at 0.5u^2 and rsqrt at 3.9u^2. tolerance_high[fn_sqrt] is 2.5, which equals the short kernel's documented bound. If the high-level dispatch fell back to the short sqrt kernel, the assertion would still pass. The fdim bound of 3.5 and the default-level sqrt bound of 3.5 are also above the documented 2.5.

Use bounds near the documented per-kernel errors, with a small margin. Allow the documented device and host differences for fp64mp2.

Review info

Run configuration

• Source mode: Auto
• Repository: NVIDIA/cccl
• Branch: main
• File: .coderabbit.yaml
• Commit: 8d33eb1
• Layers: Repository YAML
• Review profile: CHILL
• Plan: Enterprise
• Run ID: ca6f9082-6d84-49c5-991f-541d14789200

Commits

Reviewing files that changed from the base of the PR and between 8d33eb1 and 0013716.

Files selected for processing: 28

• ci/matrix.yaml
• docs/libcudacxx/extended_api/fp/fpmp.rst
• docs/libcudacxx/extended_api/fp/fpmp_error_bounds.rst
• docs/libcudacxx/extended_api/fp/fpmp_spec.rst
• libcudacxx/include/cuda/__fp/fpmp.h
• libcudacxx/include/cuda/__fp/fpmp_impl.h
• libcudacxx/include/cuda/__fp/fpmp_impl_divsqrt.h
• libcudacxx/include/cuda/__fp/fpmp_math.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_classify.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_exp.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_hyperbolic.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_lib.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_manip.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_nearint.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_pow.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_special.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_trig.h
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_cbrt.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_erfc.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_explog.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_hyperbolic.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_invtrig.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_pow.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_special_values.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_sqrt.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_trig.pass.cpp

Included review availability: This review used your included allowance. Your plan provides up to 12 included reviews per hour; 10 remain after this review.

[coderabbitai]

Major · Functional Correctness · Quick win

important: Clear the temporary override before merging.

These entries replace the normal pull_request matrix. Lines 67-69 explicitly state that this mode makes branch-protection checks fail. Remove the temporary entries after targeted validation, but retain the override: key.

[rmlarsen]

Yes: the override is there while this stack is in draft iteration and is reset to an empty override: before merging.

[coderabbitai]

Actionable comments posted: 2

Review info

Run configuration

• Source mode: Auto
• Repository: NVIDIA/cccl
• Branch: main
• File: .coderabbit.yaml
• Commit: 8d33eb1
• Layers: Repository YAML
• Review profile: CHILL
• Plan: Enterprise
• Run ID: 1c9319f4-5838-45b1-a08f-1be4da937660

Commits

Reviewing files that changed from the base of the PR and between 0013716 and 0d3de05.

Files selected for processing: 8

• docs/libcudacxx/extended_api/fp/fpmp.rst
• libcudacxx/include/cuda/__fp/fpmp_math.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_lib.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_special.h
• libcudacxx/include/cuda/__fp/fpmp_math_impl_trig.h
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_erfcx.pass.cpp
• libcudacxx/test/libcudacxx/cuda/fp/units/fpmp/math_special_values.pass.cpp

Files skipped from review as they are similar to previous changes: 2

• docs/libcudacxx/extended_api/fp/fpmp.rst
• libcudacxx/include/cuda/__fp/fpmp_math_impl.h

Included review availability: This review used your included allowance. Your plan provides up to 12 included reviews per hour; 11 remain after this review.

[rmlarsen]

/ok to test 02b1d83