import torch
import math
from typing import Optional


# Remainder solution
def _phi(j, neg_h):
    remainder = torch.zeros_like(neg_h)

    for k in range(j): 
        remainder += (neg_h)**k / math.factorial(k)
    phi_j_h = ((neg_h).exp() - remainder) / (neg_h)**j

    return phi_j_h

def calculate_gamma(c2, c3):
    return (3*(c3**3) - 2*c3) / (c2*(2 - 3*c2))

# Exact analytic solution originally calculated by Clybius. https://github.com/Clybius/ComfyUI-Extra-Samplers/tree/main
def _gamma(n: int,) -> int:
    """
    https://en.wikipedia.org/wiki/Gamma_function
    for every positive integer n,
    Γ(n) = (n-1)!
    """
    return math.factorial(n-1)

def _incomplete_gamma(s: int, x: float, gamma_s: Optional[int] = None) -> float:
    """
    https://en.wikipedia.org/wiki/Incomplete_gamma_function#Special_values
    if s is a positive integer,
    Γ(s, x) = (s-1)!*∑{k=0..s-1}(x^k/k!)
    """
    if gamma_s is None:
        gamma_s = _gamma(s)

    sum_: float = 0
    # {k=0..s-1} inclusive
    for k in range(s):
        numerator: float = x**k
        denom: int = math.factorial(k)
        quotient: float = numerator/denom
        sum_ += quotient
    incomplete_gamma_: float = sum_ * math.exp(-x) * gamma_s
    return incomplete_gamma_

def phi(j: int, neg_h: float, ):
    """
    For j={1,2,3}: you could alternatively use Kat's phi_1, phi_2, phi_3 which perform fewer steps

    Lemma 1
    https://arxiv.org/abs/2308.02157
    ϕj(-h) = 1/h^j*∫{0..h}(e^(τ-h)*(τ^(j-1))/((j-1)!)dτ)

    https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84
    = 1/h^j*[(e^(-h)*(-τ)^(-j)*τ(j))/((j-1)!)]{0..h}
    https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84+between+0+and+h
    = 1/h^j*((e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h)))/(j-1)!)
    = (e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h))/((j-1)!*h^j)
    = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/(j-1)!
    = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/Γ(j)
    = (e^(-h)*(-h)^(-j)*(1-Γ(j,-h)/Γ(j))

    requires j>0
    """
    assert j > 0
    gamma_: float = _gamma(j)
    incomp_gamma_: float = _incomplete_gamma(j, neg_h, gamma_s=gamma_)
    phi_: float = math.exp(neg_h) * neg_h**-j * (1-incomp_gamma_/gamma_)
    return phi_



from mpmath import mp, mpf, factorial, exp


mp.dps = 80   # e.g. 80 decimal digits (~ float256)

def phi_mpmath_series(j: int, neg_h: float) -> float:
    """
    Arbitrary‐precision phi_j(-h) via the remainder‐series definition,
    using mpmath’s mpf and factorial.
    """
    j = int(j)
    z = mpf(float(neg_h))    
    S = mp.mpf('0')    # S = sum_{k=0..j-1} z^k / k!
    for k in range(j):
        S += (z**k) / factorial(k)
    phi_val = (exp(z) - S) / (z**j)
    return float(phi_val)



class Phi:
    def __init__(self, h, c, analytic_solution=False): 
        self.h = h
        self.c = c
        self.cache = {}  
        if analytic_solution:
            #self.phi_f = superphi
            self.phi_f = phi_mpmath_series
            self.h = mpf(float(h))
            self.c = [mpf(c_val) for c_val in c]
            #self.c = c
            #self.phi_f = phi
        else:
            self.phi_f = phi
            #self.phi_f = _phi  # remainder method

    def __call__(self, j, i=-1):
        if (j, i) in self.cache:
            return self.cache[(j, i)]

        if i < 0:
            c = 1
        else:
            c = self.c[i - 1]
            if c == 0:
                self.cache[(j, i)] = 0
                return 0

        if j == 0 and type(c) in {float, torch.Tensor}:
            result = math.exp(float(-self.h * c))
        else:
            result = self.phi_f(j, -self.h * c)

        self.cache[(j, i)] = result

        return result



from mpmath import mp, mpf, gamma, gammainc

def superphi(j: int, neg_h: float, ):
    gamma_: float = gamma(j)
    incomp_gamma_: float = gamma_ - gammainc(j, 0, float(neg_h))
    phi_: float = float(math.exp(float(neg_h)) * neg_h**-j) * (1-incomp_gamma_/gamma_)
    return float(phi_)