





import inspect
from typing import Optional, Union

import numpy as np
import PIL
import torch
from torch.nn import functional as F
from torchvision import transforms
from transformers import CLIPFeatureExtractor, CLIPModel, CLIPTextModel, CLIPTokenizer

from diffusers import (
    AutoencoderKL,
    DDIMScheduler,
    DiffusionPipeline,
    DPMSolverMultistepScheduler,
    LMSDiscreteScheduler,
    PNDMScheduler,
    UNetaDConditionModel,
)
from diffusers.pipelines.stable_diffusion.pipeline_stable_diffusion import StableDiffusionPipelineOutput
from diffusers.utils import (
    PIL_INTERPOLATION,
    randn_tensor,
)





def lowerCAmelCase__(__snake_case       ,__snake_case       ,__snake_case							)				->      List[Any]:







	'''simple docstring'''







	if isinstance(lowerCamelCase_       ,torch.Tensor							):
		return image
	elif isinstance(lowerCamelCase_       ,PIL.Image.Image							):
		lowerCamelCase__								=		[image]

	if isinstance(image[0]       ,PIL.Image.Image							):
		lowerCamelCase__								=		[np.array(i.resize((w, h)       ,resample=PIL_INTERPOLATION['''lanczos''']							)							)[None, :] for i in image]
		lowerCamelCase__								=		np.concatenate(lowerCamelCase_       ,axis=0							)
		lowerCamelCase__								=		np.array(lowerCamelCase_							).astype(np.floataa							) / 255.0
		lowerCamelCase__								=		image.transpose(0       ,3       ,1       ,2							)
		lowerCamelCase__								=		2.0 * image - 1.0
		lowerCamelCase__								=		torch.from_numpy(lowerCamelCase_							)
	elif isinstance(image[0]       ,torch.Tensor							):
		lowerCamelCase__								=		torch.cat(lowerCamelCase_       ,dim=0							)
	return image





def lowerCAmelCase__(__snake_case       ,__snake_case       ,__snake_case       ,__snake_case=0.9_9_9_5							)				->      Tuple:







	'''simple docstring'''







	if not isinstance(lowerCamelCase_       ,np.ndarray							):
		lowerCamelCase__								=		True
		lowerCamelCase__								=		va.device
		lowerCamelCase__								=		va.cpu().numpy()
		lowerCamelCase__								=		va.cpu().numpy()

	lowerCamelCase__								=		np.sum(va * va / (np.linalg.norm(lowerCamelCase_							) * np.linalg.norm(lowerCamelCase_							))							)
	if np.abs(lowerCamelCase_							) > DOT_THRESHOLD:
		lowerCamelCase__								=		(1 - t) * va + t * va
	else:
		lowerCamelCase__								=		np.arccos(lowerCamelCase_							)
		lowerCamelCase__								=		np.sin(lowerCamelCase_							)
		lowerCamelCase__								=		theta_a * t
		lowerCamelCase__								=		np.sin(lowerCamelCase_							)
		lowerCamelCase__								=		np.sin(theta_a - theta_t							) / sin_theta_a
		lowerCamelCase__								=		sin_theta_t / sin_theta_a
		lowerCamelCase__								=		sa * va + sa * va

	if inputs_are_torch:
		lowerCamelCase__								=		torch.from_numpy(lowerCamelCase_							).to(lowerCamelCase_							)

	return va





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      List[Any]:







	'''simple docstring'''







	lowerCamelCase__								=		F.normalize(lowerCamelCase_       ,dim=-1							)
	lowerCamelCase__								=		F.normalize(lowerCamelCase_       ,dim=-1							)
	return (x - y).norm(dim=-1							).div(2							).arcsin().pow(2							).mul(2							)





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      Optional[int]:







	'''simple docstring'''







	for param in model.parameters():
		lowerCamelCase__								=		value






class 					__A    (    _a ):

			'''simple docstring'''







			def __init__(    self       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase=None       ,  __lowerCAmelCase=None       ,  __lowerCAmelCase=None       ,  ):



				'''simple docstring'''







				super().__init__()
				self.register_modules(
				    vae=__lowerCAmelCase       ,  text_encoder=__lowerCAmelCase       ,  clip_model=__lowerCAmelCase       ,  tokenizer=__lowerCAmelCase       ,  unet=__lowerCAmelCase       ,  scheduler=__lowerCAmelCase       ,  feature_extractor=__lowerCAmelCase       ,  coca_model=__lowerCAmelCase       ,  coca_tokenizer=__lowerCAmelCase       ,  coca_transform=__lowerCAmelCase       ,  )
				lowerCamelCase__								=		(
				    feature_extractor.size
				    if isinstance(feature_extractor.size       ,  __lowerCAmelCase							)
				    else feature_extractor.size['shortest_edge']
				)
				lowerCamelCase__								=		transforms.Normalize(mean=feature_extractor.image_mean       ,  std=feature_extractor.image_std							)
				set_requires_grad(self.text_encoder       ,  __lowerCAmelCase							)
				set_requires_grad(self.clip_model       ,  __lowerCAmelCase							)





			def 		__lowerCamelCase				(    self       ,  __lowerCAmelCase = "auto"							):



				'''simple docstring'''







				if slice_size == "auto":
					# half the attention head size is usually a good trade-off between
					# speed and memory
					lowerCamelCase__								=		self.unet.config.attention_head_dim // 2
				self.unet.set_attention_slice(__lowerCAmelCase							)





			def 		__lowerCamelCase				(    self							):



				'''simple docstring'''







				self.enable_attention_slicing(__lowerCAmelCase							)





			def 		__lowerCamelCase				(    self							):



				'''simple docstring'''







				set_requires_grad(self.vae       ,  __lowerCAmelCase							)





			def 		__lowerCamelCase				(    self							):



				'''simple docstring'''







				set_requires_grad(self.vae       ,  __lowerCAmelCase							)





			def 		__lowerCamelCase				(    self							):



				'''simple docstring'''







				set_requires_grad(self.unet       ,  __lowerCAmelCase							)





			def 		__lowerCamelCase				(    self							):



				'''simple docstring'''







				set_requires_grad(self.unet       ,  __lowerCAmelCase							)





			def 		__lowerCamelCase				(    self       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase							):



				'''simple docstring'''







				lowerCamelCase__								=		min(int(num_inference_steps * strength							)       ,  __lowerCAmelCase							)

				lowerCamelCase__								=		max(num_inference_steps - init_timestep       ,  0							)
				lowerCamelCase__								=		self.scheduler.timesteps[t_start:]

				return timesteps, num_inference_steps - t_start





			def 		__lowerCamelCase				(    self       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase=None							):



				'''simple docstring'''







				if not isinstance(__lowerCAmelCase       ,  torch.Tensor							):
					raise ValueError(F'`image` has to be of type `torch.Tensor` but is {type(__lowerCAmelCase							)}'							)

				lowerCamelCase__								=		image.to(device=__lowerCAmelCase       ,  dtype=__lowerCAmelCase							)

				if isinstance(__lowerCAmelCase       ,  __lowerCAmelCase							):
					lowerCamelCase__								=		[
					    self.vae.encode(image[i : i + 1]							).latent_dist.sample(generator[i]							) for i in range(__lowerCAmelCase							)
					]
					lowerCamelCase__								=		torch.cat(__lowerCAmelCase       ,  dim=0							)
				else:
					lowerCamelCase__								=		self.vae.encode(__lowerCAmelCase							).latent_dist.sample(__lowerCAmelCase							)

				# Hardcode 0.18215 because stable-diffusion-2-base has not self.vae.config.scaling_factor
				lowerCamelCase__								=		0.1_8215 * init_latents
				lowerCamelCase__								=		init_latents.repeat_interleave(__lowerCAmelCase       ,  dim=0							)

				lowerCamelCase__								=		randn_tensor(init_latents.shape       ,  generator=__lowerCAmelCase       ,  device=__lowerCAmelCase       ,  dtype=__lowerCAmelCase							)

				# get latents
				lowerCamelCase__								=		self.scheduler.add_noise(__lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase							)
				lowerCamelCase__								=		init_latents

				return latents





			def 		__lowerCamelCase				(    self       ,  __lowerCAmelCase							):



				'''simple docstring'''







				lowerCamelCase__								=		self.coca_transform(__lowerCAmelCase							).unsqueeze(0							)
				with torch.no_grad(), torch.cuda.amp.autocast():
					lowerCamelCase__								=		self.coca_model.generate(transformed_image.to(device=self.device       ,  dtype=self.coca_model.dtype							)							)
				lowerCamelCase__								=		self.coca_tokenizer.decode(generated[0].cpu().numpy()							)
				return generated.split('''<end_of_text>'''							)[0].replace('''<start_of_text>'''       ,  ''''''							).rstrip(''' .,'''							)





			def 		__lowerCamelCase				(    self       ,  __lowerCAmelCase       ,  __lowerCAmelCase							):



				'''simple docstring'''







				lowerCamelCase__								=		self.feature_extractor.preprocess(__lowerCAmelCase							)
				lowerCamelCase__								=		torch.from_numpy(clip_image_input['''pixel_values'''][0]							).unsqueeze(0							).to(self.device							).half()
				lowerCamelCase__								=		self.clip_model.get_image_features(__lowerCAmelCase							)
				lowerCamelCase__								=		image_embeddings_clip / image_embeddings_clip.norm(p=2       ,  dim=-1       ,  keepdim=__lowerCAmelCase							)
				lowerCamelCase__								=		image_embeddings_clip.repeat_interleave(__lowerCAmelCase       ,  dim=0							)
				return image_embeddings_clip





			@torch.enable_grad()
			def 		__lowerCamelCase				(    self       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  ):



				'''simple docstring'''







				lowerCamelCase__								=		latents.detach().requires_grad_()

				lowerCamelCase__								=		self.scheduler.scale_model_input(__lowerCAmelCase       ,  __lowerCAmelCase							)

				# predict the noise residual
				lowerCamelCase__								=		self.unet(__lowerCAmelCase       ,  __lowerCAmelCase       ,  encoder_hidden_states=__lowerCAmelCase							).sample

				if isinstance(self.scheduler       ,  (PNDMScheduler, DDIMScheduler, DPMSolverMultistepScheduler)							):
					lowerCamelCase__								=		self.scheduler.alphas_cumprod[timestep]
					lowerCamelCase__								=		1 - alpha_prod_t
					# compute predicted original sample from predicted noise also called
					# "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
					lowerCamelCase__								=		(latents - beta_prod_t ** 0.5 * noise_pred) / alpha_prod_t ** 0.5

					lowerCamelCase__								=		torch.sqrt(__lowerCAmelCase							)
					lowerCamelCase__								=		pred_original_sample * (fac) + latents * (1 - fac)
				elif isinstance(self.scheduler       ,  __lowerCAmelCase							):
					lowerCamelCase__								=		self.scheduler.sigmas[index]
					lowerCamelCase__								=		latents - sigma * noise_pred
				else:
					raise ValueError(F'scheduler type {type(self.scheduler							)} not supported'							)

				# Hardcode 0.18215 because stable-diffusion-2-base has not self.vae.config.scaling_factor
				lowerCamelCase__								=		1 / 0.1_8215 * sample
				lowerCamelCase__								=		self.vae.decode(__lowerCAmelCase							).sample
				lowerCamelCase__								=		(image / 2 + 0.5).clamp(0       ,  1							)

				lowerCamelCase__								=		transforms.Resize(self.feature_extractor_size							)(__lowerCAmelCase							)
				lowerCamelCase__								=		self.normalize(__lowerCAmelCase							).to(latents.dtype							)

				lowerCamelCase__								=		self.clip_model.get_image_features(__lowerCAmelCase							)
				lowerCamelCase__								=		image_embeddings_clip / image_embeddings_clip.norm(p=2       ,  dim=-1       ,  keepdim=__lowerCAmelCase							)

				lowerCamelCase__								=		spherical_dist_loss(__lowerCAmelCase       ,  __lowerCAmelCase							).mean() * clip_guidance_scale

				lowerCamelCase__								=		-torch.autograd.grad(__lowerCAmelCase       ,  __lowerCAmelCase							)[0]

				if isinstance(self.scheduler       ,  __lowerCAmelCase							):
					lowerCamelCase__								=		latents.detach() + grads * (sigma**2)
					lowerCamelCase__								=		noise_pred_original
				else:
					lowerCamelCase__								=		noise_pred_original - torch.sqrt(__lowerCAmelCase							) * grads
				return noise_pred, latents





			@torch.no_grad()
			def __call__(    self       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase = None       ,  __lowerCAmelCase = None       ,  __lowerCAmelCase = 5_1_2       ,  __lowerCAmelCase = 5_1_2       ,  __lowerCAmelCase = 0.6       ,  __lowerCAmelCase = 5_0       ,  __lowerCAmelCase = 7.5       ,  __lowerCAmelCase = 1       ,  __lowerCAmelCase = 0.0       ,  __lowerCAmelCase = 1_0_0       ,  __lowerCAmelCase = None       ,  __lowerCAmelCase = "pil"       ,  __lowerCAmelCase = True       ,  __lowerCAmelCase = 0.8       ,  __lowerCAmelCase = 0.1       ,  __lowerCAmelCase = 0.1       ,  ):



				'''simple docstring'''







				if isinstance(__lowerCAmelCase       ,  __lowerCAmelCase							) and len(__lowerCAmelCase							) != batch_size:
					raise ValueError(F'You have passed {batch_size} batch_size, but only {len(__lowerCAmelCase							)} generators.'							)

				if height % 8 != 0 or width % 8 != 0:
					raise ValueError(F'`height` and `width` have to be divisible by 8 but are {height} and {width}.'							)

				if isinstance(__lowerCAmelCase       ,  torch.Generator							) and batch_size > 1:
					lowerCamelCase__								=		[generator] + [None] * (batch_size - 1)

				lowerCamelCase__								=		[
				    ('model', self.coca_model is None),
				    ('tokenizer', self.coca_tokenizer is None),
				    ('transform', self.coca_transform is None),
				]
				lowerCamelCase__								=		[x[0] for x in coca_is_none if x[1]]
				lowerCamelCase__								=		', '.join(__lowerCAmelCase							)
				# generate prompts with coca model if prompt is None
				if content_prompt is None:
					if len(__lowerCAmelCase							):
						raise ValueError(
						    F'Content prompt is None and CoCa [{coca_is_none_str}] is None.'
						    F'Set prompt or pass Coca [{coca_is_none_str}] to DiffusionPipeline.'							)
					lowerCamelCase__								=		self.get_image_description(__lowerCAmelCase							)
				if style_prompt is None:
					if len(__lowerCAmelCase							):
						raise ValueError(
						    F'Style prompt is None and CoCa [{coca_is_none_str}] is None.'
						    F' Set prompt or pass Coca [{coca_is_none_str}] to DiffusionPipeline.'							)
					lowerCamelCase__								=		self.get_image_description(__lowerCAmelCase							)

				# get prompt text embeddings for content and style
				lowerCamelCase__								=		self.tokenizer(
				    __lowerCAmelCase       ,  padding='''max_length'''       ,  max_length=self.tokenizer.model_max_length       ,  truncation=__lowerCAmelCase       ,  return_tensors='''pt'''       ,  )
				lowerCamelCase__								=		self.text_encoder(content_text_input.input_ids.to(self.device							)							)[0]

				lowerCamelCase__								=		self.tokenizer(
				    __lowerCAmelCase       ,  padding='''max_length'''       ,  max_length=self.tokenizer.model_max_length       ,  truncation=__lowerCAmelCase       ,  return_tensors='''pt'''       ,  )
				lowerCamelCase__								=		self.text_encoder(style_text_input.input_ids.to(self.device							)							)[0]

				lowerCamelCase__								=		slerp(__lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase							)

				# duplicate text embeddings for each generation per prompt
				lowerCamelCase__								=		text_embeddings.repeat_interleave(__lowerCAmelCase       ,  dim=0							)

				# set timesteps
				lowerCamelCase__								=		'offset' in set(inspect.signature(self.scheduler.set_timesteps							).parameters.keys()							)
				lowerCamelCase__								=		{}
				if accepts_offset:
					lowerCamelCase__								=		1

				self.scheduler.set_timesteps(__lowerCAmelCase       ,  **__lowerCAmelCase							)
				# Some schedulers like PNDM have timesteps as arrays
				# It's more optimized to move all timesteps to correct device beforehand
				self.scheduler.timesteps.to(self.device							)

				lowerCamelCase__								=		self.get_timesteps(__lowerCAmelCase       ,  __lowerCAmelCase       ,  self.device							)
				lowerCamelCase__								=		timesteps[:1].repeat(__lowerCAmelCase							)

				# Preprocess image
				lowerCamelCase__								=		preprocess(__lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase							)
				lowerCamelCase__								=		self.prepare_latents(
				    __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  text_embeddings.dtype       ,  self.device       ,  __lowerCAmelCase							)

				lowerCamelCase__								=		preprocess(__lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase							)
				lowerCamelCase__								=		self.prepare_latents(
				    __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  text_embeddings.dtype       ,  self.device       ,  __lowerCAmelCase							)

				lowerCamelCase__								=		slerp(__lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase							)

				if clip_guidance_scale > 0:
					lowerCamelCase__								=		self.get_clip_image_embeddings(__lowerCAmelCase       ,  __lowerCAmelCase							)
					lowerCamelCase__								=		self.get_clip_image_embeddings(__lowerCAmelCase       ,  __lowerCAmelCase							)
					lowerCamelCase__								=		slerp(
					    __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase							)

				# here `guidance_scale` is defined analog to the guidance weight `w` of equation (2)
				# of the Imagen paper: https://arxiv.org/pdf/2205.11487.pdf . `guidance_scale = 1`
				# corresponds to doing no classifier free guidance.
				lowerCamelCase__								=		guidance_scale > 1.0
				# get unconditional embeddings for classifier free guidance
				if do_classifier_free_guidance:
					lowerCamelCase__								=		content_text_input.input_ids.shape[-1]
					lowerCamelCase__								=		self.tokenizer(['''''']       ,  padding='''max_length'''       ,  max_length=__lowerCAmelCase       ,  return_tensors='''pt'''							)
					lowerCamelCase__								=		self.text_encoder(uncond_input.input_ids.to(self.device							)							)[0]
					# duplicate unconditional embeddings for each generation per prompt
					lowerCamelCase__								=		uncond_embeddings.repeat_interleave(__lowerCAmelCase       ,  dim=0							)

					# For classifier free guidance, we need to do two forward passes.
					# Here we concatenate the unconditional and text embeddings into a single batch
					# to avoid doing two forward passes
					lowerCamelCase__								=		torch.cat([uncond_embeddings, text_embeddings]							)

				# get the initial random noise unless the user supplied it

				# Unlike in other pipelines, latents need to be generated in the target device
				# for 1-to-1 results reproducibility with the CompVis implementation.
				# However this currently doesn't work in `mps`.
				lowerCamelCase__								=		(batch_size, self.unet.config.in_channels, height // 8, width // 8)
				lowerCamelCase__								=		text_embeddings.dtype
				if latents is None:
					if self.device.type == "mps":
						# randn does not work reproducibly on mps
						lowerCamelCase__								=		torch.randn(__lowerCAmelCase       ,  generator=__lowerCAmelCase       ,  device='''cpu'''       ,  dtype=__lowerCAmelCase							).to(
						    self.device							)
					else:
						lowerCamelCase__								=		torch.randn(__lowerCAmelCase       ,  generator=__lowerCAmelCase       ,  device=self.device       ,  dtype=__lowerCAmelCase							)
				else:
					if latents.shape != latents_shape:
						raise ValueError(F'Unexpected latents shape, got {latents.shape}, expected {latents_shape}'							)
					lowerCamelCase__								=		latents.to(self.device							)

				# scale the initial noise by the standard deviation required by the scheduler
				lowerCamelCase__								=		latents * self.scheduler.init_noise_sigma

				# prepare extra kwargs for the scheduler step, since not all schedulers have the same signature
				# eta (η) is only used with the DDIMScheduler, it will be ignored for other schedulers.
				# eta corresponds to η in DDIM paper: https://arxiv.org/abs/2010.02502
				# and should be between [0, 1]
				lowerCamelCase__								=		'eta' in set(inspect.signature(self.scheduler.step							).parameters.keys()							)
				lowerCamelCase__								=		{}
				if accepts_eta:
					lowerCamelCase__								=		eta

				# check if the scheduler accepts generator
				lowerCamelCase__								=		'generator' in set(inspect.signature(self.scheduler.step							).parameters.keys()							)
				if accepts_generator:
					lowerCamelCase__								=		generator

				with self.progress_bar(total=__lowerCAmelCase							):
					for i, t in enumerate(__lowerCAmelCase							):
						# expand the latents if we are doing classifier free guidance
						lowerCamelCase__								=		torch.cat([latents] * 2							) if do_classifier_free_guidance else latents
						lowerCamelCase__								=		self.scheduler.scale_model_input(__lowerCAmelCase       ,  __lowerCAmelCase							)

						# predict the noise residual
						lowerCamelCase__								=		self.unet(__lowerCAmelCase       ,  __lowerCAmelCase       ,  encoder_hidden_states=__lowerCAmelCase							).sample

						# perform classifier free guidance
						if do_classifier_free_guidance:
							lowerCamelCase__								=		noise_pred.chunk(2							)
							lowerCamelCase__								=		noise_pred_uncond + guidance_scale * (noise_pred_text - noise_pred_uncond)

						# perform clip guidance
						if clip_guidance_scale > 0:
							lowerCamelCase__								=		(
							    text_embeddings.chunk(2							)[1] if do_classifier_free_guidance else text_embeddings
							)
							lowerCamelCase__								=		self.cond_fn(
							    __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  )

						# compute the previous noisy sample x_t -> x_t-1
						lowerCamelCase__								=		self.scheduler.step(__lowerCAmelCase       ,  __lowerCAmelCase       ,  __lowerCAmelCase       ,  **__lowerCAmelCase							).prev_sample

        # Hardcode 0.18215 because stable-diffusion-2-base has not self.vae.config.scaling_factor
				lowerCamelCase__								=		1 / 0.1_8215 * latents
				lowerCamelCase__								=		self.vae.decode(__lowerCAmelCase							).sample

				lowerCamelCase__								=		(image / 2 + 0.5).clamp(0       ,  1							)
				lowerCamelCase__								=		image.cpu().permute(0       ,  2       ,  3       ,  1							).numpy()

				if output_type == "pil":
					lowerCamelCase__								=		self.numpy_to_pil(__lowerCAmelCase							)

				if not return_dict:
					return (image, None)

				return StableDiffusionPipelineOutput(images=__lowerCAmelCase       ,  nsfw_content_detected=__lowerCAmelCase							)







from __future__ import annotations

from math import ceil, floor, sqrt



def 				UpperCamelCase_(      lowerCamelCase_ = 200_0000 )    ->						int:
    _lowercase       :							list[int]      		=		[0]
    _lowercase       :							int

    for idx in range(1							, ceil(sqrt(target * 2 ) * 1.1 ) ):
        triangle_numbers.append(triangle_numbers[-1] + idx )

    # we want this to be as close as possible to target
    _lowercase       :							int      		=		0
    # the area corresponding to the grid that gives the product closest to target
    _lowercase       :							int      		=		0
    # an estimate of b, using the quadratic formula
    _lowercase       :							float
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the triangle number corresponding to b_floor
    _lowercase       :							int
    # the triangle number corresponding to b_ceil
    _lowercase       :							int

    for idx_a, triangle_a in enumerate(triangle_numbers[1:]							, 1 ):
        _lowercase       :							Optional[int]      		=		(-1 + sqrt(1 + 8 * target / triangle_a )) / 2
        _lowercase       :							List[str]      		=		floor(lowerCamelCase_ )
        _lowercase       :							Dict      		=		ceil(lowerCamelCase_ )
        _lowercase       :							List[str]      		=		triangle_numbers[b_floor]
        _lowercase       :							List[str]      		=		triangle_numbers[b_ceil]

        if abs(target - triangle_b_first_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Union[str, Any]      		=		triangle_b_first_guess * triangle_a
            _lowercase       :							Union[str, Any]      		=		idx_a * b_floor

        if abs(target - triangle_b_second_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Any      		=		triangle_b_second_guess * triangle_a
            _lowercase       :							Optional[Any]      		=		idx_a * b_ceil

    return area


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


