l j dZddlmZgdZddlZddlmZddlmZddl m Z m Z m Z ddl Z d#d$dZedd%dZGdde Ze deZd&d Zd'd"ZdS)(z!A collection of simple functions.) annotations) binary_searchchooseclipsigmoidN)Callable) lru_cache)AnyProtocolTypeVar-C6?functionCallable[[float], float]targetfloat lower_bound upper_bound tolerancereturn float | Nonec|}|}tjtj||g}t||z |krtjtj||g}fd|||fD\}} } ||kr|S| |kr|S||cxkr| krnn | |kr|}n|}n||cxkr| krnn||}}ndSt||z |k|S)aSearches for a value in a range by repeatedly dividing the range in half. To be more precise, performs numerical binary search to determine the input to ``function``, between the bounds given, that outputs ``target`` to within ``tolerance`` (default of 0.0001). Returns ``None`` if no input can be found within the bounds. Examples -------- Consider the polynomial :math:`x^2 + 3x + 1` where we search for a target value of :math:`11`. An exact solution is :math:`x = 2`. :: >>> solution = binary_search(lambda x: x**2 + 3*x + 1, 11, 0, 5) >>> bool(abs(solution - 2) < 1e-4) True >>> solution = binary_search(lambda x: x**2 + 3*x + 1, 11, 0, 5, tolerance=0.01) >>> bool(abs(solution - 2) < 0.01) True Searching in the interval :math:`[0, 5]` for a target value of :math:`71` does not yield a solution:: >>> binary_search(lambda x: x**2 + 3*x + 1, 71, 0, 5) is None True c3.K|]}|VdSN).0hrs W/home/agentuser/manim-venv/lib/python3.11/site-packages/manim/utils/simple_functions.py z binary_search..<s+88ahhqkk888888N)npmeanarrayabs) rrrrrlhrhmhlxmxrxs ` rrrs)F B B"b**++B b2g,, " " WRXr2h'' ( (8888BB<888 B <<I <<I     2     F{{ &    2     BB4! b2g,, " "$ Ir )maxsizenintkc0tj||}|S)a3The binomial coefficient n choose k. :math:`\binom{n}{k}` describes the number of possible choices of :math:`k` elements from a set of :math:`n` elements. References ---------- - https://en.wikipedia.org/wiki/Combination - https://docs.python.org/3/library/math.html#math.comb )mathcomb)r,r.values rrrOs1aE LrceZdZddZddZdS) Comparableotherr rboolcdSrrselfr5s r__lt__zComparable.__lt__`rcdSrrr8s r__gt__zComparable.__gt__br;rN)r5r rr6)__name__ __module__ __qualname__r:r=rrrr4r4_s(----------rr4 ComparableT)boundamin_amax_ac&||kr|S||kr|S|S)aoClips ``a`` to the interval [``min_a``, ``max_a``]. Accepts any comparable objects (i.e. those that support <, >). Returns ``a`` if it is between ``min_a`` and ``max_a``. Otherwise, whichever of ``min_a`` and ``max_a`` is closest. Examples -------- :: >>> clip(15, 11, 20) 15 >>> clip('a', 'h', 'k') 'h' r)rCrDrEs rrrhs%  5yy U Hrxc<ddtj| zz }|S)a/Returns the output of the logistic function. The logistic function, a common example of a sigmoid function, is defined as :math:`\frac{1}{1 + e^{-x}}`. References ---------- - https://en.wikipedia.org/wiki/Sigmoid_function - https://en.wikipedia.org/wiki/Logistic_function g?)r exp)rGr2s rrrs!!bfaRjj.)E Lr)r ) rrrrrrrrrrrr)r,r-r.r-rr-)rCrArDrArErArrA)rGrrr)__doc__ __future__r__all__r0collections.abcr functoolsr typingr r r numpyr rrr4rArrrrrrRsF''""""""    $$$$$$)))))))))) 88888v 2    ........ gm:666     .      r