Diophantine reduction for exp, sin, cos, tan in the fixed module

[fredrik-johansson]

This reimplements arb's Diophantine argument reduction for elementary functions in the fixed module along with larger lookup tables allowing one to use more primes than arb's default n = 13, currently up to n = 48.

• fixed_exp_diophantine - exp on [0, 1) by reduction against logarithms of the first n primes.
• fixed_sin_cos_diophantine, fixed_tan_diophantine - the trigonometric counterpart using Gaussian primes.
• relation tables (fixed_rel_table) - precomputed for n = 2, 4, 6, 8, 10, 12, 13, 16, 20, 24, 32, 40, 48 in both the rational and Gaussian cases; any other n up to 64 is generated on first use and cached per thread. arb's 13-prime tables are removed in favour of these
• Machin formulas (fixed_machin_table) - arb's sets, plus new ones for n = 32, 40, 48 of each kind and every intermediate size up to 32, taken from fredrik-johansson/machin. The coefficient matrices are not stored: they are reconstructed on first use by factoring the arguments and inverting the exponent matrix (<1 ms, under 1% of an evaluation at 10⁵ bits and above), which keeps the table file at ~50 KB instead of ~275 KB.
• arb_log_primes_vec_bsplit / arb_atan_gauss_primes_vec_bsplit now use the shared tables, so they gain the larger sets and the intermediate sizes; their previous hand-written size dispatch and 32-bit-only fallback sets are gone.
• examples/machin.c additionally verifies some of FLINT's tabulated formulas exactly with ca

Developed using Claude Fable 5.1 and Opus 5.

Baselines

Absolute time to evaluate the elementary functions without any precomputations:

precision	exp_notab	sin_cos_notab	sin_cos_notab / exp_notab
16k bits (256 limbs)	0.0003 s	0.0004 s	1.19
64k bits (1024 limbs)	0.0023 s	0.0027 s	1.19
256k bits (4096 limbs)	0.0147 s	0.0184 s	1.25
1M bits (16384 limbs)	0.0900 s	0.1061 s	1.18
4M bits (65536 limbs)	0.5123 s	0.6352 s	1.24

exp

Timings (precomp / warm) are relative to computing the exponential function with the table-free fixed_exp_notab. So 15.22 / 0.246 means that a first call requires 15x as much time as a table-free evaluation for precomputations, while subsequent evaluations with a warm cache are 4x faster than the table-free evaluation.

precision	bitwise	diophantine, 13	20	32	48
16k	15.22 / 0.246	2.88 / 0.439	3.92 / 0.398	6.63 / 0.364	10.60 / 0.433
64k	23.57 / 0.297	2.58 / 0.577	3.52 / 0.512	5.41 / 0.453	8.49 / 0.426
256k	26.93 / 0.320	2.20 / 0.627	3.07 / 0.508	4.66 / 0.460	7.08 / 0.431
1M	33.01 / 0.384	1.87 / 0.591	2.60 / 0.550	3.97 / 0.493	6.02 / 0.481
4M	38.66 / 0.462	1.71 / 0.702	2.38 / 0.644	3.58 / 0.626	5.64 / 0.630

sin_cos

precision	bitwise	diophantine, 13	20	32	48
16k	10.33 / 0.487	3.06 / 0.697	4.61 / 0.630	6.69 / 0.573	10.96 / 0.528
64k	14.48 / 0.467	2.80 / 0.743	3.91 / 0.679	5.85 / 0.529	8.81 / 0.573
256k	19.84 / 0.448	2.45 / 0.732	3.41 / 0.688	4.98 / 0.607	7.65 / 0.553
1M	25.67 / 0.517	2.04 / 0.779	2.91 / 0.709	4.23 / 0.641	6.42 / 0.603
4M	33.37 / 0.627	1.94 / 0.874	2.73 / 0.813	3.90 / 0.732	5.91 / 0.695

Break-even

How many times one needs to call the function for the precomputation to pay off vs no precomputation at all.

exp

precision	bitwise	diophantine, 13	diophantine, 32
16k	20	5	10
64k	34	6	10
256k	40	6	9
1M	54	5	8
4M	72	6	10

sin_cos

precision	bitwise	diophantine, 13	diophantine, 32
16k	15	6	11
64k	20	6	9
256k	25	5	8
1M	39	5	8
4M	54	5	8

bitwise against diophantine (32 primes)

precision	exp	sin_cos
16k	73	42
64k	116	139
256k	159	93
1M	266	173
4M	214	281

Note: fixed_exp_diophantine and fixed_sin_cos_diophantine are 1.2-1.6x faster than arb_exp and arb_sin_cos when using arb's default n=13.