CG solver: Replace PRP+ with Hager-Zhang update
Replace the Polak-Ribière-Plus (PRP+) conjugate direction update with the Hager-Zhang formula in `mj_solPrimal`.
While this change has negligible effect under float64, it leads to a significant 17.5% throughput speedup over PRP+ under float32 (measured via `engine_cg_convergence_test`). This performance gain is driven by:
* A 9.4% reduction in CG iterations per step.
* A 12.1% reduction in line search evaluations per step.
The full output of the comparison is
```
================================================================
1/4: HZ + float64
================================================================
CG Convergence: 2humanoid100.xml
1000 Newton steps, 100 evaluation points
nv = 654, nq = 756
metric: ||qacc_cg - qacc_newton|| / ||qacc_newton||
Warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 3.8532e-01 | 1.4285e+00 | 5.00 | 22368
10 | 2.0809e-01 | 9.9315e-01 | 10.00 | 44591
20 | 6.4977e-02 | 2.3410e-01 | 20.00 | 89417
40 | 5.2839e-03 | 2.1916e-02 | 40.00 | 180984
80 | 3.9709e-05 | 3.0010e-04 | 80.00 | 364241
160 | 3.2498e-09 | 3.5480e-08 | 160.00 | 730131
-------+-------------+-------------+------------+---------
No warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 9.0740e-01 | 5.0045e+00 | 5.00 | 22123
10 | 3.9506e-01 | 2.1927e+00 | 10.00 | 44879
20 | 1.2496e-01 | 6.8911e-01 | 20.00 | 90341
40 | 1.2846e-02 | 6.9533e-02 | 40.00 | 182869
80 | 1.0288e-04 | 8.6457e-04 | 80.00 | 368312
160 | 5.9820e-09 | 5.0903e-08 | 160.00 | 738479
-------+-------------+-------------+------------+---------
Tolerance sweep (iterations = 100, warmstart):
Tol | Mean Err | Max Err | Mean Iters | Max Iters | Solver us | LS evals
-----------+-------------+-------------+------------+------------+-------------+---------
1e-04 | 1.1297e-02 | 8.9331e-02 | 34.88 | 51 | 131002.94 | 12052
1e-06 | 1.0907e-03 | 7.7468e-03 | 50.87 | 72 | 184172.44 | 16700
1e-08 | 1.1485e-04 | 1.0085e-03 | 66.82 | 91 | 232189.34 | 20194
1e-10 | 1.1357e-05 | 8.1547e-05 | 81.97 | 100 | 276114.42 | 23264
1e-12 | 3.7695e-06 | 2.8592e-05 | 90.47 | 100 | 299526.58 | 24964
0 | 3.5019e-06 | 2.8592e-05 | 100.00 | 100 | 2062208.36 | 456276
-----------+-------------+-------------+------------+------------+-------------+---------
Total solver time: 3185214.08 us, avg time per iter: 74.9445 us
Pipeline mode (consecutive mj_step, tolerance = 1e-8):
1000 steps, nv = 654
Steps/s : 380
us/step (total) : 2630.0
us/step (constr) : 2103.7 (80.0%)
CG iters/step : 63.28
LS evals/step : 190.75
us/iter : 33.24
================================================================
2/4: PRP+ + float64
================================================================
CG Convergence: 2humanoid100.xml
1000 Newton steps, 100 evaluation points
nv = 654, nq = 756
metric: ||qacc_cg - qacc_newton|| / ||qacc_newton||
Warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 3.8533e-01 | 1.4285e+00 | 5.00 | 22228
10 | 2.0808e-01 | 9.9315e-01 | 10.00 | 44873
20 | 6.4978e-02 | 2.3410e-01 | 20.00 | 89349
40 | 5.2895e-03 | 2.1916e-02 | 40.00 | 179827
80 | 4.0188e-05 | 3.0010e-04 | 80.00 | 363740
160 | 3.2891e-09 | 3.5480e-08 | 160.00 | 733905
-------+-------------+-------------+------------+---------
No warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 9.0740e-01 | 5.0045e+00 | 5.00 | 22590
10 | 3.9506e-01 | 2.1927e+00 | 10.00 | 45093
20 | 1.2496e-01 | 6.8911e-01 | 20.00 | 90393
40 | 1.2846e-02 | 6.9533e-02 | 40.00 | 182422
80 | 1.0288e-04 | 8.6457e-04 | 80.00 | 367264
160 | 5.9811e-09 | 5.0903e-08 | 160.00 | 739556
-------+-------------+-------------+------------+---------
Tolerance sweep (iterations = 100, warmstart):
Tol | Mean Err | Max Err | Mean Iters | Max Iters | Solver us | LS evals
-----------+-------------+-------------+------------+------------+-------------+---------
1e-04 | 1.1325e-02 | 8.9941e-02 | 34.95 | 52 | 129456.81 | 12060
1e-06 | 1.0910e-03 | 7.7469e-03 | 50.85 | 72 | 181004.51 | 16687
1e-08 | 1.1488e-04 | 1.0085e-03 | 66.83 | 91 | 228183.13 | 20189
1e-10 | 1.1401e-05 | 8.1547e-05 | 81.98 | 100 | 269635.40 | 23259
1e-12 | 3.8040e-06 | 2.8592e-05 | 90.52 | 100 | 293811.25 | 24967
0 | 3.5543e-06 | 2.8592e-05 | 100.00 | 100 | 2051927.47 | 456000
-----------+-------------+-------------+------------+------------+-------------+---------
Total solver time: 3154018.57 us, avg time per iter: 74.1895 us
Pipeline mode (consecutive mj_step, tolerance = 1e-8):
1000 steps, nv = 654
Steps/s : 382
us/step (total) : 2616.6
us/step (constr) : 2091.8 (79.9%)
CG iters/step : 63.57
LS evals/step : 193.35
us/iter : 32.91
================================================================
3/4: HZ + float32
================================================================
CG Convergence: 2humanoid100.xml
1000 Newton steps, 100 evaluation points
nv = 654, nq = 756
metric: ||qacc_cg - qacc_newton|| / ||qacc_newton||
Warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 3.6748e-01 | 1.2281e+00 | 5.00 | 19239
10 | 2.0034e-01 | 6.8689e-01 | 10.00 | 39001
20 | 6.1972e-02 | 1.9859e-01 | 20.00 | 80112
40 | 4.0704e-03 | 1.5797e-02 | 40.00 | 168211
80 | 2.4380e-05 | 1.5803e-04 | 79.45 | 347735
160 | 7.7413e-07 | 4.4732e-06 | 157.27 | 704210
-------+-------------+-------------+------------+---------
No warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 9.3385e-01 | 4.9648e+00 | 5.00 | 19272
10 | 3.6117e-01 | 1.9618e+00 | 10.00 | 37982
20 | 9.7490e-02 | 5.1451e-01 | 20.00 | 76502
40 | 9.5924e-03 | 5.5088e-02 | 40.00 | 162782
80 | 6.7689e-05 | 5.1113e-04 | 79.91 | 344281
160 | 1.5541e-06 | 7.6955e-06 | 157.30 | 697076
-------+-------------+-------------+------------+---------
Tolerance sweep (iterations = 100, warmstart):
Tol | Mean Err | Max Err | Mean Iters | Max Iters | Solver us | LS evals
-----------+-------------+-------------+------------+------------+-------------+---------
1e-04 | 1.5891e-02 | 1.2929e-01 | 32.56 | 50 | 136254.00 | 10980
1e-06 | 1.5719e-03 | 1.0990e-02 | 47.42 | 68 | 193566.00 | 15418
1e-08 | 1.6260e-04 | 1.0680e-03 | 62.70 | 86 | 252661.00 | 20162
1e-10 | 1.5664e-05 | 1.0999e-04 | 78.10 | 100 | 311535.00 | 24509
1e-12 | 2.8152e-06 | 1.5236e-05 | 90.39 | 100 | 356814.00 | 27902
0 | 2.4989e-06 | 1.5025e-05 | 99.05 | 100 | 1980580.00 | 436825
-----------+-------------+-------------+------------+------------+-------------+---------
Total solver time: 3231410.00 us, avg time per iter: 78.7726 us
Pipeline mode (consecutive mj_step, tolerance = 1e-8):
1000 steps, nv = 654
Steps/s : 349
us/step (total) : 2862.7
us/step (constr) : 2379.3 (83.1%)
CG iters/step : 61.97
LS evals/step : 199.01
us/iter : 38.39
================================================================
4/4: PRP+ + float32
================================================================
CG Convergence: 2humanoid100.xml
1000 Newton steps, 100 evaluation points
nv = 654, nq = 756
metric: ||qacc_cg - qacc_newton|| / ||qacc_newton||
Warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 3.6753e-01 | 1.2281e+00 | 5.00 | 19052
10 | 2.0028e-01 | 6.8689e-01 | 10.00 | 38761
20 | 6.1737e-02 | 1.9863e-01 | 20.00 | 79937
40 | 4.0915e-03 | 1.5821e-02 | 39.99 | 168159
80 | 2.4360e-05 | 1.5808e-04 | 79.59 | 348385
160 | 7.8367e-07 | 4.1549e-06 | 157.04 | 700760
-------+-------------+-------------+------------+---------
No warmstart (tolerance = 0):
Iters | Mean Err | Max Err | Mean Iters | LS evals
-------+-------------+-------------+------------+---------
5 | 9.3385e-01 | 4.9648e+00 | 5.00 | 19494
10 | 3.6117e-01 | 1.9618e+00 | 10.00 | 38480
20 | 9.7485e-02 | 5.1451e-01 | 20.00 | 77213
40 | 9.5904e-03 | 5.5078e-02 | 40.00 | 162472
80 | 6.7647e-05 | 5.1087e-04 | 79.87 | 342990
160 | 1.4979e-06 | 7.2291e-06 | 157.92 | 699052
-------+-------------+-------------+------------+---------
Tolerance sweep (iterations = 100, warmstart):
Tol | Mean Err | Max Err | Mean Iters | Max Iters | Solver us | LS evals
-----------+-------------+-------------+------------+------------+-------------+---------
1e-04 | 1.5998e-02 | 1.3172e-01 | 32.49 | 50 | 135111.00 | 10953
1e-06 | 1.5878e-03 | 1.1052e-02 | 47.33 | 67 | 190914.00 | 15418
1e-08 | 1.5901e-04 | 1.0680e-03 | 62.65 | 84 | 250609.00 | 20215
1e-10 | 1.5953e-05 | 1.1116e-04 | 78.01 | 100 | 307067.00 | 24562
1e-12 | 2.8520e-06 | 1.5174e-05 | 90.27 | 100 | 353010.00 | 27931
0 | 2.5161e-06 | 1.5129e-05 | 99.39 | 100 | 1999478.00 | 438790
-----------+-------------+-------------+------------+------------+-------------+---------
Total solver time: 3236189.00 us, avg time per iter: 78.9045 us
Pipeline mode (consecutive mj_step, tolerance = 1e-8):
1000 steps, nv = 654
Steps/s : 298
us/step (total) : 3352.8
us/step (constr) : 2832.0 (84.5%)
CG iters/step : 68.43
LS evals/step : 226.33
us/iter : 41.39
```
PiperOrigin-RevId: 928590104
Change-Id: I1a96730f50f444d6141d8978feb3519009daf320
Yuval Tassa
828052e6f4