diff --git a/manimlib/utils/bezier.py b/manimlib/utils/bezier.py
index 26000972..51b124cf 100644
--- a/manimlib/utils/bezier.py
+++ b/manimlib/utils/bezier.py
@@ -15,9 +15,7 @@ if TYPE_CHECKING:
     from typing import Callable, Sequence, TypeVar
     from manimlib.typing import VectN, FloatArray
 
-    T = TypeVar("T")
-
-    Scalable = TypeVar("Scalable", float, VectN)
+    Scalable = TypeVar("Scalable", float, FloatArray)
 
 
 CLOSED_THRESHOLD = 0.001
diff --git a/manimlib/utils/simple_functions.py b/manimlib/utils/simple_functions.py
index 143bf350..1236b44a 100644
--- a/manimlib/utils/simple_functions.py
+++ b/manimlib/utils/simple_functions.py
@@ -5,26 +5,34 @@ import math
 
 import numpy as np
 
+from typing import TYPE_CHECKING
+if TYPE_CHECKING:
+    from typing import Callable, TypeVar
+    from manimlib.typing import FloatArray
 
-def sigmoid(x):
+    Scalable = TypeVar("Scalable", float, FloatArray)
+
+
+
+def sigmoid(x: float | FloatArray):
     return 1.0 / (1 + np.exp(-x))
 
 
 @lru_cache(maxsize=10)
-def choose(n, k):
+def choose(n: int, k: int) -> int:
     return math.comb(n, k)
 
 
-def gen_choose(n, r):
-    return np.prod(np.arange(n, n - r, -1)) / math.factorial(r)
+def gen_choose(n: int, r: int) -> int:
+    return int(np.prod(range(n, n - r, -1)) / math.factorial(r))
 
 
-def get_num_args(function):
+def get_num_args(function: Callable) -> int:
     return len(get_parameters(function))
 
 
-def get_parameters(function):
-    return inspect.signature(function).parameters
+def get_parameters(function: Callable) -> list:
+    return list(inspect.signature(function).parameters.keys())
 
 # Just to have a less heavyweight name for this extremely common operation
 #
@@ -33,7 +41,7 @@ def get_parameters(function):
 # but for now, we just allow the option to handle indeterminate 0/0.
 
 
-def clip(a, min_a, max_a):
+def clip(a: float, min_a: float, max_a: float) -> float:
     if a < min_a:
         return min_a
     elif a > max_a:
@@ -41,7 +49,7 @@ def clip(a, min_a, max_a):
     return a
 
 
-def fdiv(a, b, zero_over_zero_value=None):
+def fdiv(a: Scalable, b: Scalable, zero_over_zero_value: Scalable | None = None) -> Scalable:
     if zero_over_zero_value is not None:
         out = np.full_like(a, zero_over_zero_value)
         where = np.logical_or(a != 0, b != 0)
@@ -52,15 +60,15 @@ def fdiv(a, b, zero_over_zero_value=None):
     return np.true_divide(a, b, out=out, where=where)
 
 
-def binary_search(function,
-                  target,
-                  lower_bound,
-                  upper_bound,
-                  tolerance=1e-4):
+def binary_search(function: Callable[[float], float],
+                  target: float,
+                  lower_bound: float,
+                  upper_bound: float,
+                  tolerance:float = 1e-4) -> float | None:
     lh = lower_bound
     rh = upper_bound
+    mh = (lh + rh) / 2
     while abs(rh - lh) > tolerance:
-        mh = np.mean([lh, rh])
         lx, mx, rx = [function(h) for h in (lh, mh, rh)]
         if lx == target:
             return lx
@@ -76,10 +84,11 @@ def binary_search(function,
             lh, rh = rh, lh
         else:
             return None
+        mh = (lh + rh) / 2
     return mh
 
 
-def hash_string(string):
+def hash_string(string: str) -> str:
     # Truncating at 16 bytes for cleanliness
     hasher = hashlib.sha256(string.encode())
     return hasher.hexdigest()[:16]