Optimize double-word integer factorization

[fredrik-johansson]

This fixes most of the slowness of fmpz_factor in the 65-160 bit range observed in #2625. The speedup is largest around 80 bits where we gain nearly 6x on average for random integers. Performance is also slightly improved for 3-4 word integers.

Several changes:

• Lower SQUFOF threshold to 72 bits
• Optimize Pollard-Brent rho and ECM for double-word arithmetic and use these in appropriate ranges
• Reduce initialization costs for qsieve
• Optimize removal for double-word values in qsieve
• Tweak tuning values for qsieve
• Optimize modular square root in qsieve

Average time per factorization to factor 100 random integers of the given bit size:

          uniformly random                 semiprimes
bits        old        new  speedup         old        new  speedup
  72  2.740e-04  1.187e-04  2.310x    2.102e-03  2.218e-03  0.948x
  80  8.527e-04  1.431e-04  5.959x    3.445e-03  2.625e-03  1.312x
  88  1.559e-03  3.535e-04  4.411x    2.697e-03  1.923e-03  1.402x
  96  2.387e-03  6.841e-04  3.489x    3.678e-03  2.745e-03  1.340x
 104  2.642e-03  1.053e-03  2.509x    5.304e-03  4.094e-03  1.296x
 112  3.377e-03  1.354e-03  2.495x    8.032e-03  6.359e-03  1.263x
 120  4.202e-03  2.161e-03  1.945x    1.308e-02  1.163e-02  1.125x
 128  6.392e-03  3.862e-03  1.655x    2.366e-02  1.741e-02  1.359x
 136  9.211e-03  6.904e-03  1.334x    3.489e-02  2.818e-02  1.238x
 144  1.471e-02  1.190e-02  1.236x    6.179e-02  5.237e-02  1.180x
 152  1.703e-02  1.479e-02  1.152x    9.785e-02  8.795e-02  1.113x
 160  2.768e-02  2.527e-02  1.095x    2.010e-01  1.511e-01  1.330x
 168  4.596e-02  4.154e-02  1.106x    3.128e-01  2.504e-01  1.249x
 176  5.872e-02  5.031e-02  1.167x    5.764e-01  4.122e-01  1.398x
 184  9.741e-02  7.924e-02  1.229x    9.098e-01  6.880e-01  1.323x
 192  1.359e-01  1.170e-01  1.162x    1.504e+00  1.356e+00  1.109x

Worst-case time for the same 100 inputs:

          uniformly random                 semiprimes
bits        old        new  speedup         old        new  speedup
  72  5.800e-03  4.600e-03  1.261x    1.900e-02  1.900e-02  1.000x
  80  6.700e-03  2.100e-03  3.190x    6.300e-03  5.000e-03  1.260x
  88  1.000e-02  4.100e-03  2.439x    4.000e-03  3.400e-03  1.176x
  96  7.200e-03  3.700e-03  1.946x    4.400e-03  3.500e-03  1.257x
 104  7.200e-03  5.300e-03  1.358x    7.800e-03  6.500e-03  1.200x
 112  1.000e-02  8.100e-03  1.235x    1.100e-02  8.900e-03  1.236x
 120  1.500e-02  1.400e-02  1.071x    2.400e-02  2.100e-02  1.143x
 128  2.400e-02  2.100e-02  1.143x    4.000e-02  3.100e-02  1.290x
 136  4.100e-02  3.700e-02  1.108x    6.400e-02  5.500e-02  1.164x
 144  6.000e-02  5.400e-02  1.111x    2.160e-01  1.980e-01  1.091x
 152  1.020e-01  9.700e-02  1.052x    1.670e-01  1.560e-01  1.071x
 160  1.900e-01  1.820e-01  1.044x    3.770e-01  2.590e-01  1.456x
 168  3.140e-01  2.550e-01  1.231x    5.810e-01  4.990e-01  1.164x
 176  6.070e-01  4.480e-01  1.355x    1.020e+00  7.500e-01  1.360x
 184  1.248e+00  8.900e-01  1.402x    1.398e+00  2.649e+00  0.528x
 192  1.730e+00  1.576e+00  1.098x    3.318e+00  3.079e+00  1.078x

Tried before to get Opus 4.8 to do this, but it persistently failed to identify the real bottlenecks/solutions; Fable refused to touch this problem (work on integer factorization apparently triggers its safety guardrails); finally Opus 5 got the job done in a few hours.

[fredrik-johansson]

Opus also identified a trivial improvement to fmpz_is_prime in the Pocklington and Morrison tests. Time to run fmpz_is_prime on 10 random primes of each bit size:

bits        old       new   speedup
  80  2.960e-05  2.990e-05  0.990x
  88  5.230e-03  4.880e-03  1.072x
  96  5.690e-03  5.350e-03  1.064x
 104  1.130e-02  1.060e-02  1.066x
 112  1.210e-02  1.140e-02  1.061x
 120  1.570e-02  1.470e-02  1.068x
 128  1.900e-02  1.790e-02  1.061x
 136  2.700e-02  2.510e-02  1.076x
 144  2.490e-02  2.340e-02  1.064x
 152  5.120e-02  3.580e-02  1.430x
 160  4.310e-02  3.850e-02  1.119x
 168  8.140e-02  4.580e-02  1.777x
 176  1.030e-01  4.220e-02  2.441x
 184  1.010e-01  4.770e-02  2.117x
 192  9.100e-02  5.740e-02  1.585x
 200  1.200e-01  6.980e-02  1.719x
 208  1.230e-01  6.740e-02  1.825x
 216  1.340e-01  7.700e-02  1.740x
 224  1.320e-01  8.480e-02  1.557x
 232  1.400e-01  8.800e-02  1.591x
 240  1.960e-01  1.100e-01  1.782x
 248  1.560e-01  1.140e-01  1.368x
 256  2.120e-01  1.240e-01  1.710x
 264  1.820e-01  1.450e-01  1.255x
 272  2.040e-01  1.360e-01  1.500x
 280  2.770e-01  1.610e-01  1.720x
 288  2.680e-01  1.740e-01  1.540x
 296  2.690e-01  1.840e-01  1.462x
 304  2.850e-01  1.950e-01  1.462x
 312  3.030e-01  2.070e-01  1.464x
 320  3.210e-01  2.240e-01  1.433x
 328  2.950e-01  2.540e-01  1.161x
 336  3.470e-01  2.690e-01  1.290x
 344  3.700e-01  2.850e-01  1.298x
 352  4.230e-01  3.350e-01  1.263x
 360  4.920e-01  3.750e-01  1.312x
 368  4.770e-01  3.590e-01  1.329x
 376  5.250e-01  4.170e-01  1.259x
 384  5.240e-01  4.320e-01  1.213x
 392  5.690e-01  4.730e-01  1.203x
 400  6.240e-01  5.220e-01  1.195x
 408  6.190e-01  5.270e-01  1.175x
 416  6.350e-01  5.480e-01  1.159x
 424  7.110e-01  5.770e-01  1.232x
 432  7.480e-01  6.280e-01  1.191x
 440  7.760e-01  6.590e-01  1.178x
 448  8.270e-01  7.060e-01  1.171x
 456  8.730e-01  7.590e-01  1.150x
 464  9.230e-01  7.980e-01  1.157x
 472  9.260e-01  7.960e-01  1.163x
 480  9.600e-01  8.520e-01  1.127x
 488  9.960e-01  9.020e-01  1.104x
 496  1.058e+00  9.180e-01  1.153x
 504  1.068e+00  9.540e-01  1.119x
 512  1.134e+00  1.014e+00  1.118x