





import math







def 					_SCREAMING_SNAKE_CASE							(  a    ) -> bool:
					__A      :		Optional[int]         =							math.loga(math.sqrt(4 * positive_integer + 1    ) / 2 + 1 / 2    )

					return exponent == int(UpperCamelCase_    )







def 					_SCREAMING_SNAKE_CASE							(  a = 1 / 1_23_45    ) -> int:
					__A      :		Union[str, Any]         =							0
					__A      :		int         =							0

					__A      :		List[Any]         =							3
					while True:
										__A      :		Union[str, Any]         =							(integer**2 - 1) / 4
										# if candidate is an integer, then there is a partition for k
										if partition_candidate == int(UpperCamelCase_    ):
															__A      :		Optional[Any]         =							int(UpperCamelCase_    )
															total_partitions += 1
															if check_partition_perfect(UpperCamelCase_    ):
																				perfect_partitions += 1
										if perfect_partitions > 0:
															if perfect_partitions / total_partitions < max_proportion:
																				return int(UpperCamelCase_    )
										integer += 1


if __name__ == "__main__":
				print(F"""{solution() = }""")









import math
import os
import re
import sys
import unittest
from pathlib import Path
from typing import Tuple
from unittest.mock import patch

from parameterized import parameterized

from transformers.testing_utils import (
    CaptureStderr,
    ExtendSysPath,
    TestCasePlus,
    execute_subprocess_async,
    get_gpu_count,
    get_torch_dist_unique_port,
    require_apex,
    require_bitsandbytes,
    require_fairscale,
    require_torch,
    require_torch_gpu,
    require_torch_multi_gpu,
    require_torch_non_multi_gpu,
    slow,
)
from transformers.trainer_callback import TrainerState
from transformers.trainer_utils import set_seed


_lowercase    	=		os.path.abspath(os.path.dirname(__file__))
with ExtendSysPath(F"{bindir}/../../examples/pytorch/translation"):
   from run_translation import main  # noqa


set_seed(42)
_lowercase    	=		"sshleifer/student_marian_en_ro_6_1"
_lowercase    	=		"sshleifer/tiny-mbart"



@require_torch
class      _UpperCAmelCase      (							A__  ):




     def 		snake_case_		(					self			,    a__=False			,    a__=None			,    a__=True			,    a__=True			,    a__=True			,    a__=True			,    ):
         A__					=       self.run_trainer(
             eval_steps=1			,    max_len=1_2			,    model_name=a__			,    num_train_epochs=1			,    distributed=a__			,    extra_args_str=a__			,    predict_with_generate=a__			,    do_train=a__			,    do_eval=a__			,    do_predict=a__			,    )
         A__					=       TrainerState.load_from_json(os.path.join(a__			,    '''trainer_state.json''')).log_history

         if not do_eval:
             return

         A__					=       [log for log in logs if '''eval_loss''' in log.keys()]

         A__					=       eval_metrics[0]
         if predict_with_generate:
             assert "eval_bleu" in first_step_stats

             A__					=       eval_metrics[-1]
             assert isinstance(last_step_stats['''eval_bleu''']			,    a__)
             assert not math.isnan(float(last_step_stats['''eval_loss'''])), "eval_loss must not be `nan`"




     @require_torch_non_multi_gpu
     def 		snake_case_		(					self):
         self.run_seqaseq_quick()




     @require_torch_multi_gpu
     def 		snake_case_		(					self):
         self.run_seqaseq_quick(distributed=a__)




     @require_torch_multi_gpu
     def 		snake_case_		(					self):
         self.run_seqaseq_quick(distributed=a__)




     @unittest.skip('''Requires an update of the env running those tests''')
     @require_torch_multi_gpu
     @require_fairscale
     def 		snake_case_		(					self):
         self.run_seqaseq_quick(distributed=a__			,    extra_args_str='''--sharded_ddp simple''')




     @unittest.skip('''Requires an update of the env running those tests''')
     @require_torch_multi_gpu
     @require_fairscale
     def 		snake_case_		(					self):
         self.run_seqaseq_quick(distributed=a__			,    extra_args_str='''--sharded_ddp simple --fp16''')




     @unittest.skip('''Requires an update of the env running those tests''')
     @require_torch_multi_gpu
     @require_fairscale
     def 		snake_case_		(					self):
         self.run_seqaseq_quick(distributed=a__			,    extra_args_str='''--sharded_ddp zero_dp_2'''			,    predict_with_generate=a__)




     @unittest.skip('''Requires an update of the env running those tests''')
     @require_torch_multi_gpu
     @require_fairscale
     def 		snake_case_		(					self):
         self.run_seqaseq_quick(
             distributed=a__			,    extra_args_str='''--sharded_ddp zero_dp_2 --fp16'''			,    predict_with_generate=a__)




     @require_apex
     @require_torch_gpu
     def 		snake_case_		(					self):
         # XXX: apex breaks the trainer if it's run twice e.g. run_seq2seq.main() from the same
         # program and it breaks other tests that run from the same pytest worker, therefore until this is
         # sorted out it must be run only in an external program, that is distributed=True in this
         # test and only under one or more gpus - if we want cpu will need to make a special test
         #
         # specifically to the problem traced it to self.optimizer.step() - if it's run 2nd time via
         # 2nd main() call it botches the future eval.
         #
         self.run_seqaseq_quick(distributed=a__			,    extra_args_str='''--fp16 --fp16_backend=apex''')
         # test 2nd time - was getting eval_loss': nan'
         # to reproduce the problem set distributed=False
         self.run_seqaseq_quick(distributed=a__			,    extra_args_str='''--fp16 --fp16_backend=apex''')




     @parameterized.expand(['''base''', '''low''', '''high''', '''mixed'''])
     @require_torch_multi_gpu
     def 		snake_case_		(					self			,    a__):
         # as each sub-test is slow-ish split into multiple sub-tests to avoid CI timeout
         A__					=       {
             # test with the default log_level - should be info and thus log info once
             '''base''': {'''extra_args_str''': '''''', '''n_matches''': 1},
             # test with low log_level and log_level_replica - should be noisy on all processes
             # now the info string should appear twice on 2 processes
             '''low''': {'''extra_args_str''': '''--log_level debug --log_level_replica debug''', '''n_matches''': 2},
             # test with high log_level and low log_level_replica
             # now the info string should appear once only on the replica
             '''high''': {'''extra_args_str''': '''--log_level error --log_level_replica debug''', '''n_matches''': 1},
             # test with high log_level and log_level_replica - should be quiet on all processes
             '''mixed''': {'''extra_args_str''': '''--log_level error --log_level_replica error''', '''n_matches''': 0},
         }

         A__					=       experiments[experiment_id]
         A__					=       {'''distributed''': True, '''predict_with_generate''': False, '''do_eval''': False, '''do_predict''': False}
         A__					=       '''Running training'''
         with CaptureStderr() as cl:
             self.run_seqaseq_quick(**a__			,    extra_args_str=data['''extra_args_str'''])
         A__					=       len(re.findall(a__			,    cl.err))
         self.assertEqual(a__			,    data['''n_matches'''])




     @slow
     def 		snake_case_		(					self):
         A__					=       self.run_trainer(
             eval_steps=2			,    max_len=1_2_8			,    model_name=a__			,    learning_rate=3e-4			,    num_train_epochs=1_0			,    distributed=a__			,    )

         # Check metrics
         A__					=       TrainerState.load_from_json(os.path.join(a__			,    '''trainer_state.json''')).log_history
         A__					=       [log for log in logs if '''eval_loss''' in log.keys()]
         A__					=       eval_metrics[0]
         A__					=       eval_metrics[-1]

         assert first_step_stats["eval_loss"] > last_step_stats["eval_loss"], "model learned nothing"
         assert isinstance(last_step_stats['''eval_bleu''']			,    a__)

         # test if do_predict saves generations and metrics
         A__					=       os.listdir(a__)
         A__					=       {os.path.basename(a__) for p in contents}
         assert "generated_predictions.txt" in contents
         assert "predict_results.json" in contents




     @slow
     @require_bitsandbytes
     def 		snake_case_		(					self):
         from transformers.training_args import OptimizerNames

         def train_and_return_metrics(a__) -> Tuple[int, float]:
             A__					=       '''--skip_memory_metrics 0'''

             A__					=       self.run_trainer(
                 max_len=1_2_8			,    model_name=a__			,    learning_rate=3e-4			,    num_train_epochs=1			,    optim=a__			,    distributed=a__			,    extra_args_str=a__			,    do_eval=a__			,    do_predict=a__			,    n_gpus_to_use=1			,    )

             # Check metrics
             A__					=       TrainerState.load_from_json(Path(a__			,    '''trainer_state.json''')).log_history
             A__					=       int(logs[0]['''train_mem_gpu_peaked_delta'''] / 2**2_0)
             A__					=       int(logs[0]['''train_mem_gpu_alloc_delta'''] / 2**2_0)

             A__					=       logs[0]['''train_loss''']
             return gpu_peak_mem_mb, gpu_alloc_mem_mb, loss

         A__			,						A__			,						A__					=       train_and_return_metrics(OptimizerNames.ADAMW_TORCH.value)
         A__			,						A__			,						A__					=       train_and_return_metrics(OptimizerNames.ADAMW_BNB.value)

         A__					=       gpu_alloc_mem_orig - gpu_alloc_mem_bnb

         A__					=       gpu_peak_mem_orig + gpu_alloc_mem_orig
         A__					=       gpu_peak_mem_bnb + gpu_alloc_mem_bnb
         A__					=       gpu_total_mem_orig - gpu_total_mem_bnb

         # sshleifer/student_marian_en_ro_6_1 has 54M parameter, 29M of which is `nn.Embedding` which
         # doesn't get quantized and remains in fp32. Therefore we only have 25M parameters quantized
         # in 2 bytes and the diff in optim memory usage is derived as so:
         #
         # - normal 25*8=~200MB (8 bytes per param)
         # - bnb    25*2= ~50MB (2 bytes per param)
         #
         # Thus we should expect ~150MB total memory saved.
         #
         # Peak memory should be the same - the total should be different by about that same margin
         #
         # After leaving a small margin to accommodate for differences between gpus let's check
         # that we have at least 120MB in savings
         A__					=       1_2_0

         # uncomment the following if this test starts failing - requires py38 for a new print feature
         # gpu_peak_mem_diff = gpu_peak_mem_orig - gpu_peak_mem_bnb
         # print(f"{gpu_alloc_mem_orig=}MB {gpu_peak_mem_orig=}MB {gpu_alloc_mem_orig+gpu_peak_mem_orig=}MB")
         # print(f" {gpu_alloc_mem_bnb=}MB  {gpu_peak_mem_bnb=}MB  {gpu_alloc_mem_bnb+gpu_peak_mem_bnb=}MB")
         # print(f"{gpu_alloc_mem_diff=}MB")
         # print(f"{gpu_peak_mem_diff=}MB")
         # print(f"{gpu_total_mem_orig=}MB, {gpu_total_mem_bnb=}MB")
         # print(f"{gpu_total_mem_diff=}MB, {gpu_total_mem_diff=}MB")

         self.assertGreater(
             a__			,    a__			,    '''should use ~150MB less alloc gpu memory with BNB, compared to without it for this model but got'''
             F" a difference of {gpu_alloc_mem_diff}MB, with gpu_alloc_mem_orig={gpu_alloc_mem_orig}MB and"
             F" gpu_alloc_mem_bnb={gpu_alloc_mem_bnb}MB"			,    )

         self.assertGreater(
             a__			,    a__			,    '''should use ~150MB less total gpu memory with BNB, compared to without it for this model but got'''
             F" a difference of {gpu_total_mem_diff}MB, with gpu_total_mem_orig={gpu_total_mem_orig}MB and"
             F" gpu_total_mem_bnb={gpu_total_mem_bnb}MB"			,    )

         self.assertEqual(
             a__			,    a__			,    F"loss should be the same, but got loss_orig={loss_orig}, loss_bnb={loss_bnb}")




     def 		snake_case_		(					self			,    a__			,    a__			,    a__			,    a__ = 3e-3			,    a__ = "adafactor"			,    a__ = False			,    a__ = None			,    a__ = 0			,    a__ = True			,    a__ = True			,    a__ = True			,    a__ = True			,    a__ = None			,    ):
         A__					=       self.test_file_dir / '''../fixtures/tests_samples/wmt_en_ro'''
         A__					=       self.get_auto_remove_tmp_dir()
         A__					=       F"\n            --model_name_or_path {model_name}\n            --train_file {data_dir}/train.json\n            --validation_file {data_dir}/val.json\n            --test_file {data_dir}/test.json\n            --output_dir {output_dir}\n            --overwrite_output_dir\n            --max_train_samples 8\n            --max_source_length {max_len}\n            --max_target_length {max_len}\n            --do_train\n            --num_train_epochs {str(a__)}\n            --per_device_train_batch_size 4\n            --learning_rate {learning_rate}\n            --warmup_steps 8\n            --logging_steps 0\n            --logging_strategy no\n            --save_steps {str(a__)}\n            --group_by_length\n            --label_smoothing_factor 0.1\n            --target_lang ro_RO\n            --source_lang en_XX\n        ".split()

         A__					=       F"\n            --do_eval\n            --per_device_eval_batch_size 4\n            --max_eval_samples 8\n            --val_max_target_length {max_len}\n            --evaluation_strategy steps\n            --eval_steps {str(a__)}\n        ".split()

         A__					=       '''
            --do_predict
        '''.split()

         A__					=       []
         if do_train:
             args += args_train

         if do_eval:
             args += args_eval

         if do_predict:
             args += args_predict

         if predict_with_generate:
             args += "--predict_with_generate".split()

         if do_train:
             if optim == "adafactor":
                 args += "--adafactor".split()
             else:
                 args += F"--optim {optim}".split()

         if extra_args_str is not None:
             args += extra_args_str.split()

         if distributed:
             if n_gpus_to_use is None:
                 A__					=       get_gpu_count()
             A__					=       get_torch_dist_unique_port()
             A__					=       F"\n                -m torch.distributed.run\n                --nproc_per_node={n_gpus_to_use}\n                --master_port={master_port}\n                {self.examples_dir_str}/pytorch/translation/run_translation.py\n            ".split()
             A__					=       [sys.executable] + distributed_args + args
             # keep for quick debug
             # print(" ".join([f"\nPYTHONPATH={self.src_dir_str}"] +cmd)); die
             execute_subprocess_async(a__			,    env=self.get_env())
         else:
             A__					=       ['''run_translation.py'''] + args
             with patch.object(a__			,    '''argv'''			,    a__):
                 main()

         return output_dir
