gistfile1.txt

import math
import numpy as np

def f(t, v):
    return v * (.5**t)

def g(t, v):
    return v * (1 -t/1.442)

def h(t):
    return .5 ** t


ln2 = 0.6931471805599453
def taylor_half_power_t5(t: float) -> float:
    """
    Approximate (1/2)**t by the Taylor series for exp(-t ln 2),
    truncated after the t^5 term:

      (1/2)**t ≈ ∑_{k=0..5} [(-t·ln2)^k / k!]

    Parameters
    ----------
    t : float
        The exponent.

    Returns
    -------
    float
        5th-order Taylor approximation to (1/2)**t.
    """

    # sum terms k = 0,1,2,3,4,5
    return sum(((-t * ln2) ** k) / math.factorial(k) for k in range(6))

    
nframes = 20
framesep = 1/nframes

ts = np.random.uniform(0, 1, nframes)
ts /= np.sum(ts)

v = 1.
for t in ts:
    v = f(t, v)
print(t, v)

v = 1.
for t in ts:
    v = g(t, v)
print(t, v)

v = 1.
for t in ts:
    v = v * h(t)
print(t, v)

v = 1.
for t in ts:
    v = v * taylor_half_power_t5(t)
print(t, v)