'''simple docstring'''

# DISCLAIMER: This file is strongly influenced by https://github.com/ermongroup/ddim

from dataclasses import dataclass
from typing import Optional, Tuple, Union

import flax
import jax
import jax.numpy as jnp

from ..configuration_utils import ConfigMixin, register_to_config
from .scheduling_utils_flax import (
    CommonSchedulerState,
    FlaxKarrasDiffusionSchedulers,
    FlaxSchedulerMixin,
    FlaxSchedulerOutput,
    add_noise_common,
    get_velocity_common,
)



@flax.struct.dataclass
class 			UpperCAmelCase  :





       '''simple docstring'''



       SCREAMING_SNAKE_CASE							=    42

       # setable values
       SCREAMING_SNAKE_CASE							=    42
       SCREAMING_SNAKE_CASE							=    42
       SCREAMING_SNAKE_CASE							=    None

       @classmethod
       def _lowerCAmelCase(						cls	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase )  ->  Any:
        return cls(common=__lowerCAmelCase	,   init_noise_sigma=__lowerCAmelCase	,   timesteps=__lowerCAmelCase )



@dataclass
class 			UpperCAmelCase  (						a__					):





       '''simple docstring'''



       SCREAMING_SNAKE_CASE							=    42



class 			UpperCAmelCase  (						a__    ,						a__					):





       '''simple docstring'''



       SCREAMING_SNAKE_CASE							=    [e.name for e in FlaxKarrasDiffusionSchedulers]

       SCREAMING_SNAKE_CASE							=    42

       @property
       def _lowerCAmelCase(						self )  ->  str:
        return True

       @register_to_config
       def __init__(						self	,   __lowerCAmelCase = 1000	,   __lowerCAmelCase = 0.0_0_0_1	,   __lowerCAmelCase = 0.0_2	,   __lowerCAmelCase = "linear"	,   __lowerCAmelCase = None	,   __lowerCAmelCase = "fixed_small"	,   __lowerCAmelCase = True	,   __lowerCAmelCase = "epsilon"	,   __lowerCAmelCase = jnp.floataa	,   )  ->  Union[str, Any]:
        lowercase__				:    Dict  =					dtype

       def _lowerCAmelCase(						self	,   __lowerCAmelCase = None )  ->  DDPMSchedulerState:
        if common is None:
         lowercase__				:    List[Any]  =					CommonSchedulerState.create(self )

        # standard deviation of the initial noise distribution
        lowercase__				:    Optional[Any]  =					jnp.array(1.0	,   dtype=self.dtype )

        lowercase__				:    Any  =					jnp.arange(0	,   self.config.num_train_timesteps ).round()[::-1]

        return DDPMSchedulerState.create(
            common=__lowerCAmelCase	,   init_noise_sigma=__lowerCAmelCase	,   timesteps=__lowerCAmelCase	,   )

       def _lowerCAmelCase(						self	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase = None )  ->  jnp.ndarray:
        return sample

       def _lowerCAmelCase(						self	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase = () )  ->  DDPMSchedulerState:
        lowercase__				:    Any  =					self.config.num_train_timesteps // num_inference_steps
        # creates integer timesteps by multiplying by ratio
        # rounding to avoid issues when num_inference_step is power of 3
        lowercase__				:    int  =					(jnp.arange(0	,   __lowerCAmelCase ) * step_ratio).round()[::-1]

        return state.replace(
            num_inference_steps=__lowerCAmelCase	,   timesteps=__lowerCAmelCase	,   )

       def _lowerCAmelCase(						self	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase=None	,   __lowerCAmelCase=None )  ->  Dict:
        lowercase__				:    str  =					state.common.alphas_cumprod[t]
        lowercase__				:    Any  =					jnp.where(t > 0	,   state.common.alphas_cumprod[t - 1]	,   jnp.array(1.0	,   dtype=self.dtype ) )

        # For t > 0, compute predicted variance βt (see formula (6) and (7) from https://arxiv.org/pdf/2006.11239.pdf)
        # and sample from it to get previous sample
        # x_{t-1} ~ N(pred_prev_sample, variance) == add variance to pred_sample
        lowercase__				:    Dict  =					(1 - alpha_prod_t_prev) / (1 - alpha_prod_t) * state.common.betas[t]

        if variance_type is None:
         lowercase__				:    Optional[int]  =					self.config.variance_type

        # hacks - were probably added for training stability
        if variance_type == "fixed_small":
         lowercase__				:    int  =					jnp.clip(__lowerCAmelCase	,   a_min=1E-20 )
        # for rl-diffuser https://arxiv.org/abs/2205.09991
        elif variance_type == "fixed_small_log":
         lowercase__				:    List[str]  =					jnp.log(jnp.clip(__lowerCAmelCase	,   a_min=1E-20 ) )
        elif variance_type == "fixed_large":
         lowercase__				:    List[str]  =					state.common.betas[t]
        elif variance_type == "fixed_large_log":
         # Glide max_log
         lowercase__				:    Optional[int]  =					jnp.log(state.common.betas[t] )
        elif variance_type == "learned":
         return predicted_variance
        elif variance_type == "learned_range":
         lowercase__				:    Any  =					variance
         lowercase__				:    Any  =					state.common.betas[t]
         lowercase__				:    Tuple  =					(predicted_variance + 1) / 2
         lowercase__				:    Tuple  =					frac * max_log + (1 - frac) * min_log

        return variance

       def _lowerCAmelCase(						self	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase = None	,   __lowerCAmelCase = True	,   )  ->  Union[FlaxDDPMSchedulerOutput, Tuple]:
        lowercase__				:    str  =					timestep

        if key is None:
         lowercase__				:    Optional[int]  =					jax.random.PRNGKey(0 )

        if model_output.shape[1] == sample.shape[1] * 2 and self.config.variance_type in ["learned", "learned_range"]:
         lowercase__ ,							lowercase__				:    Tuple  =					jnp.split(__lowerCAmelCase	,   sample.shape[1]	,   axis=1 )
        else:
         lowercase__				:    Any  =					None

        # 1. compute alphas, betas
        lowercase__				:    List[Any]  =					state.common.alphas_cumprod[t]
        lowercase__				:    Tuple  =					jnp.where(t > 0	,   state.common.alphas_cumprod[t - 1]	,   jnp.array(1.0	,   dtype=self.dtype ) )
        lowercase__				:    Union[str, Any]  =					1 - alpha_prod_t
        lowercase__				:    Optional[Any]  =					1 - alpha_prod_t_prev

        # 2. compute predicted original sample from predicted noise also called
        # "predicted x_0" of formula (15) from https://arxiv.org/pdf/2006.11239.pdf
        if self.config.prediction_type == "epsilon":
         lowercase__				:    Optional[Any]  =					(sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5
        elif self.config.prediction_type == "sample":
         lowercase__				:    Tuple  =					model_output
        elif self.config.prediction_type == "v_prediction":
         lowercase__				:    Tuple  =					(alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
        else:
         raise ValueError(
             F"""prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample` """
             ''' for the FlaxDDPMScheduler.''' )

        # 3. Clip "predicted x_0"
        if self.config.clip_sample:
         lowercase__				:    List[str]  =					jnp.clip(__lowerCAmelCase	,   -1	,   1 )

        # 4. Compute coefficients for pred_original_sample x_0 and current sample x_t
        # See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
        lowercase__				:    Dict  =					(alpha_prod_t_prev ** 0.5 * state.common.betas[t]) / beta_prod_t
        lowercase__				:    Union[str, Any]  =					state.common.alphas[t] ** 0.5 * beta_prod_t_prev / beta_prod_t

        # 5. Compute predicted previous sample µ_t
        # See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
        lowercase__				:    Dict  =					pred_original_sample_coeff * pred_original_sample + current_sample_coeff * sample

        # 6. Add noise
        def random_variance():
         lowercase__				:    Dict  =					jax.random.split(__lowerCAmelCase	,   num=1 )
         lowercase__				:    Dict  =					jax.random.normal(__lowerCAmelCase	,   shape=model_output.shape	,   dtype=self.dtype )
         return (self._get_variance(__lowerCAmelCase	,   __lowerCAmelCase	,   predicted_variance=__lowerCAmelCase ) ** 0.5) * noise

        lowercase__				:    Any  =					jnp.where(t > 0	,   random_variance()	,   jnp.zeros(model_output.shape	,   dtype=self.dtype ) )

        lowercase__				:    int  =					pred_prev_sample + variance

        if not return_dict:
         return (pred_prev_sample, state)

        return FlaxDDPMSchedulerOutput(prev_sample=__lowerCAmelCase	,   state=__lowerCAmelCase )

       def _lowerCAmelCase(						self	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase	,   )  ->  jnp.ndarray:
        return add_noise_common(state.common	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase )

       def _lowerCAmelCase(						self	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase	,   )  ->  jnp.ndarray:
        return get_velocity_common(state.common	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase )


       def __len__(						self )  ->  Dict:
        return self.config.num_train_timesteps


'''simple docstring'''

def 			__UpperCamelCase       (   UpperCAmelCase     ):
 lowercase__				:    Optional[int]  =					1
 for i in range(1		,     num + 1     ):
  fact *= i
 return fact
def 			__UpperCamelCase       (   UpperCAmelCase     ):
 lowercase__				:    Optional[Any]  =					0
 while number > 0:
  lowercase__				:    str  =					number % 10
  sum_of_digits += last_digit
  lowercase__				:    List[str]  =					number // 10  # Removing the last_digit from the given number
 return sum_of_digits
def 			__UpperCamelCase       (   UpperCAmelCase = 100     ):
 lowercase__				:    Optional[int]  =					factorial(UpperCAmelCase     )
 lowercase__				:    Dict  =					split_and_add(UpperCAmelCase     )
 return result


if __name__ == "__main__":
  print(solution(int(input("""Enter the Number: """).strip())))

