venv/lib/python3.11/site-packages/sympy/simplify/gammasimp.py

from sympy.core import Function, S, Mul, Pow, Add
from sympy.core.sorting import ordered, default_sort_key
from sympy.core.function import expand_func
from sympy.core.symbol import Dummy
from sympy.functions import gamma, sqrt, sin
from sympy.polys import factor, cancel
from sympy.utilities.iterables import sift, uniq


def gammasimp(expr):
    r"""
    Simplify expressions with gamma functions.

    Explanation
    ===========

    This function takes as input an expression containing gamma
    functions or functions that can be rewritten in terms of gamma
    functions and tries to minimize the number of those functions and
    reduce the size of their arguments.

    The algorithm works by rewriting all gamma functions as expressions
    involving rising factorials (Pochhammer symbols) and applies
    recurrence relations and other transformations applicable to rising
    factorials, to reduce their arguments, possibly letting the resulting
    rising factorial to cancel. Rising factorials with the second argument
    being an integer are expanded into polynomial forms and finally all
    other rising factorial are rewritten in terms of gamma functions.

    Then the following two steps are performed.

    1. Reduce the number of gammas by applying the reflection theorem
       gamma(x)*gamma(1-x) == pi/sin(pi*x).
    2. Reduce the number of gammas by applying the multiplication theorem
       gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).

    It then reduces the number of prefactors by absorbing them into gammas
    where possible and expands gammas with rational argument.

    All transformation rules can be found (or were derived from) here:

    .. [1] https://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/
    .. [2] https://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/

    Examples
    ========

    >>> from sympy.simplify import gammasimp
    >>> from sympy import gamma, Symbol
    >>> from sympy.abc import x
    >>> n = Symbol('n', integer = True)

    >>> gammasimp(gamma(x)/gamma(x - 3))
    (x - 3)*(x - 2)*(x - 1)
    >>> gammasimp(gamma(n + 3))
    gamma(n + 3)

    """

    expr = expr.rewrite(gamma)

    # compute_ST will be looking for Functions and we don't want
    # it looking for non-gamma functions: issue 22606
    # so we mask free, non-gamma functions
    f = expr.atoms(Function)
    # take out gammas
    gammas = {i for i in f if isinstance(i, gamma)}
    if not gammas:
        return expr  # avoid side effects like factoring
    f -= gammas
    # keep only those without bound symbols
    f = f & expr.as_dummy().atoms(Function)
    if f:
        dum, fun, simp = zip(*[
            (Dummy(), fi, fi.func(*[
                _gammasimp(a, as_comb=False) for a in fi.args]))
            for fi in ordered(f)])
        d = expr.xreplace(dict(zip(fun, dum)))
        return _gammasimp(d, as_comb=False).xreplace(dict(zip(dum, simp)))

    return _gammasimp(expr, as_comb=False)


def _gammasimp(expr, as_comb):
    """
    Helper function for gammasimp and combsimp.

    Explanation
    ===========

    Simplifies expressions written in terms of gamma function. If
    as_comb is True, it tries to preserve integer arguments. See
    docstring of gammasimp for more information. This was part of
    combsimp() in combsimp.py.
    """
    expr = expr.replace(gamma,
        lambda n: _rf(1, (n - 1).expand()))

    if as_comb:
        expr = expr.replace(_rf,
            lambda a, b: gamma(b + 1))
    else:
        expr = expr.replace(_rf,
            lambda a, b: gamma(a + b)/gamma(a))

    def rule_gamma(expr, level=0):
        """ Simplify products of gamma functions further. """

        if expr.is_Atom:
            return expr

        def gamma_rat(x):
            # helper to simplify ratios of gammas
            was = x.count(gamma)
            xx = x.replace(gamma, lambda n: _rf(1, (n - 1).expand()
                ).replace(_rf, lambda a, b: gamma(a + b)/gamma(a)))
            if xx.count(gamma) < was:
                x = xx
            return x

        def gamma_factor(x):
            # return True if there is a gamma factor in shallow args
            if isinstance(x, gamma):
                return True
            if x.is_Add or x.is_Mul:
                return any(gamma_factor(xi) for xi in x.args)
            if x.is_Pow and (x.exp.is_integer or x.base.is_positive):
                return gamma_factor(x.base)
            return False

        # recursion step
        if level == 0:
            expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args])
            level += 1

        if not expr.is_Mul:
            return expr

        # non-commutative step
        if level == 1:
            args, nc = expr.args_cnc()
            if not args:
                return expr
            if nc:
                return rule_gamma(Mul._from_args(args), level + 1)*Mul._from_args(nc)
            level += 1

        # pure gamma handling, not factor absorption
        if level == 2:
            T, F = sift(expr.args, gamma_factor, binary=True)
            gamma_ind = Mul(*F)
            d = Mul(*T)

            nd, dd = d.as_numer_denom()
            for ipass in range(2):
                args = list(ordered(Mul.make_args(nd)))
                for i, ni in enumerate(args):
                    if ni.is_Add:
                        ni, dd = Add(*[
                            rule_gamma(gamma_rat(a/dd), level + 1) for a in ni.args]
                            ).as_numer_denom()
                        args[i] = ni
                        if not dd.has(gamma):
                            break
                nd = Mul(*args)
                if ipass ==  0 and not gamma_factor(nd):
                    break
                nd, dd = dd, nd  # now process in reversed order
            expr = gamma_ind*nd/dd
            if not (expr.is_Mul and (gamma_factor(dd) or gamma_factor(nd))):
                return expr
            level += 1

        # iteration until constant
        if level == 3:
            while True:
                was = expr
                expr = rule_gamma(expr, 4)
                if expr == was:
                    return expr

        numer_gammas = []
        denom_gammas = []
        numer_others = []
        denom_others = []
        def explicate(p):
            if p is S.One:
                return None, []
            b, e = p.as_base_exp()
            if e.is_Integer:
                if isinstance(b, gamma):
                    return True, [b.args[0]]*e
                else:
                    return False, [b]*e
            else:
                return False, [p]

        newargs = list(ordered(expr.args))
        while newargs:
            n, d = newargs.pop().as_numer_denom()
            isg, l = explicate(n)
            if isg:
                numer_gammas.extend(l)
            elif isg is False:
                numer_others.extend(l)
            isg, l = explicate(d)
            if isg:
                denom_gammas.extend(l)
            elif isg is False:
                denom_others.extend(l)

        # =========== level 2 work: pure gamma manipulation =========

        if not as_comb:
            # Try to reduce the number of gamma factors by applying the
            # reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x)
            for gammas, numer, denom in [(
                numer_gammas, numer_others, denom_others),
                    (denom_gammas, denom_others, numer_others)]:
                new = []
                while gammas:
                    g1 = gammas.pop()
                    if g1.is_integer:
                        new.append(g1)
                        continue
                    for i, g2 in enumerate(gammas):
                        n = g1 + g2 - 1
                        if not n.is_Integer:
                            continue
                        numer.append(S.Pi)
                        denom.append(sin(S.Pi*g1))
                        gammas.pop(i)
                        if n > 0:
                            numer.extend(1 - g1 + k for k in range(n))
                        elif n < 0:
                            denom.extend(-g1 - k for k in range(-n))
                        break
                    else:
                        new.append(g1)
                # /!\ updating IN PLACE
                gammas[:] = new

            # Try to reduce the number of gammas by using the duplication
            # theorem to cancel an upper and lower: gamma(2*s)/gamma(s) =
            # 2**(2*s + 1)/(4*sqrt(pi))*gamma(s + 1/2). Although this could
            # be done with higher argument ratios like gamma(3*x)/gamma(x),
            # this would not reduce the number of gammas as in this case.
            for ng, dg, no, do in [(numer_gammas, denom_gammas, numer_others,
                                    denom_others),
                                   (denom_gammas, numer_gammas, denom_others,
                                    numer_others)]:

                while True:
                    for x in ng:
                        for y in dg:
                            n = x - 2*y
                            if n.is_Integer:
                                break
                        else:
                            continue
                        break
                    else:
                        break
                    ng.remove(x)
                    dg.remove(y)
                    if n > 0:
                        no.extend(2*y + k for k in range(n))
                    elif n < 0:
                        do.extend(2*y - 1 - k for k in range(-n))
                    ng.append(y + S.Half)
                    no.append(2**(2*y - 1))
                    do.append(sqrt(S.Pi))

            # Try to reduce the number of gamma factors by applying the
            # multiplication theorem (used when n gammas with args differing
            # by 1/n mod 1 are encountered).
            #
            # run of 2 with args differing by 1/2
            #
            # >>> gammasimp(gamma(x)*gamma(x+S.Half))
            # 2*sqrt(2)*2**(-2*x - 1/2)*sqrt(pi)*gamma(2*x)
            #
            # run of 3 args differing by 1/3 (mod 1)
            #
            # >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(2)/3))
            # 6*3**(-3*x - 1/2)*pi*gamma(3*x)
            # >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(5)/3))
            # 2*3**(-3*x - 1/2)*pi*(3*x + 2)*gamma(3*x)
            #
            def _run(coeffs):
                # find runs in coeffs such that the difference in terms (mod 1)
                # of t1, t2, ..., tn is 1/n
                u = list(uniq(coeffs))
                for i in range(len(u)):
                    dj = ([((u[j] - u[i]) % 1, j) for j in range(i + 1, len(u))])
                    for one, j in dj:
                        if one.p == 1 and one.q != 1:
                            n = one.q
                            got = [i]
                            get = list(range(1, n))
                            for d, j in dj:
                                m = n*d
                                if m.is_Integer and m in get:
                                    get.remove(m)
                                    got.append(j)
                                    if not get:
                                        break
                            else:
                                continue
                            for i, j in enumerate(got):
                                c = u[j]
                                coeffs.remove(c)
                                got[i] = c
                            return one.q, got[0], got[1:]

            def _mult_thm(gammas, numer, denom):
                # pull off and analyze the leading coefficient from each gamma arg
                # looking for runs in those Rationals

                # expr -> coeff + resid -> rats[resid] = coeff
                rats = {}
                for g in gammas:
                    c, resid = g.as_coeff_Add()
                    rats.setdefault(resid, []).append(c)

                # look for runs in Rationals for each resid
                keys = sorted(rats, key=default_sort_key)
                for resid in keys:
                    coeffs = sorted(rats[resid])
                    new = []
                    while True:
                        run = _run(coeffs)
                        if run is None:
                            break

                        # process the sequence that was found:
                        # 1) convert all the gamma functions to have the right
                        #    argument (could be off by an integer)
                        # 2) append the factors corresponding to the theorem
                        # 3) append the new gamma function

                        n, ui, other = run

                        # (1)
                        for u in other:
                            con = resid + u - 1
                            for k in range(int(u - ui)):
                                numer.append(con - k)

                        con = n*(resid + ui)  # for (2) and (3)

                        # (2)
                        numer.append((2*S.Pi)**(S(n - 1)/2)*
                                     n**(S.Half - con))
                        # (3)
                        new.append(con)

                    # restore resid to coeffs
                    rats[resid] = [resid + c for c in coeffs] + new

                # rebuild the gamma arguments
                g = []
                for resid in keys:
                    g += rats[resid]
                # /!\ updating IN PLACE
                gammas[:] = g

            for l, numer, denom in [(numer_gammas, numer_others, denom_others),
                                    (denom_gammas, denom_others, numer_others)]:
                _mult_thm(l, numer, denom)

        # =========== level >= 2 work: factor absorption =========

        if level >= 2:
            # Try to absorb factors into the gammas: x*gamma(x) -> gamma(x + 1)
            # and gamma(x)/(x - 1) -> gamma(x - 1)
            # This code (in particular repeated calls to find_fuzzy) can be very
            # slow.
            def find_fuzzy(l, x):
                if not l:
                    return
                S1, T1 = compute_ST(x)
                for y in l:
                    S2, T2 = inv[y]
                    if T1 != T2 or (not S1.intersection(S2) and
                                    (S1 != set() or S2 != set())):
                        continue
                    # XXX we want some simplification (e.g. cancel or
                    # simplify) but no matter what it's slow.
                    a = len(cancel(x/y).free_symbols)
                    b = len(x.free_symbols)
                    c = len(y.free_symbols)
                    # TODO is there a better heuristic?
                    if a == 0 and (b > 0 or c > 0):
                        return y

            # We thus try to avoid expensive calls by building the following
            # "invariants": For every factor or gamma function argument
            #   - the set of free symbols S
            #   - the set of functional components T
            # We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset
            # or S1 == S2 == emptyset)
            inv = {}

            def compute_ST(expr):
                if expr in inv:
                    return inv[expr]
                return (expr.free_symbols, expr.atoms(Function).union(
                        {e.exp for e in expr.atoms(Pow)}))

            def update_ST(expr):
                inv[expr] = compute_ST(expr)
            for expr in numer_gammas + denom_gammas + numer_others + denom_others:
                update_ST(expr)

            for gammas, numer, denom in [(
                numer_gammas, numer_others, denom_others),
                    (denom_gammas, denom_others, numer_others)]:
                new = []
                while gammas:
                    g = gammas.pop()
                    cont = True
                    while cont:
                        cont = False
                        y = find_fuzzy(numer, g)
                        if y is not None:
                            numer.remove(y)
                            if y != g:
                                numer.append(y/g)
                                update_ST(y/g)
                            g += 1
                            cont = True
                        y = find_fuzzy(denom, g - 1)
                        if y is not None:
                            denom.remove(y)
                            if y != g - 1:
                                numer.append((g - 1)/y)
                                update_ST((g - 1)/y)
                            g -= 1
                            cont = True
                    new.append(g)
                # /!\ updating IN PLACE
                gammas[:] = new

        # =========== rebuild expr ==================================

        return Mul(*[gamma(g) for g in numer_gammas]) \
            / Mul(*[gamma(g) for g in denom_gammas]) \
            * Mul(*numer_others) / Mul(*denom_others)

    was = factor(expr)
    # (for some reason we cannot use Basic.replace in this case)
    expr = rule_gamma(was)
    if expr != was:
        expr = factor(expr)

    expr = expr.replace(gamma,
        lambda n: expand_func(gamma(n)) if n.is_Rational else gamma(n))

    return expr


class _rf(Function):
    @classmethod
    def eval(cls, a, b):
        if b.is_Integer:
            if not b:
                return S.One

            n = int(b)

            if n > 0:
                return Mul(*[a + i for i in range(n)])
            elif n < 0:
                return 1/Mul(*[a - i for i in range(1, -n + 1)])
        else:
            if b.is_Add:
                c, _b = b.as_coeff_Add()

                if c.is_Integer:
                    if c > 0:
                        return _rf(a, _b)*_rf(a + _b, c)
                    elif c < 0:
                        return _rf(a, _b)/_rf(a + _b + c, -c)

            if a.is_Add:
                c, _a = a.as_coeff_Add()

                if c.is_Integer:
                    if c > 0:
                        return _rf(_a, b)*_rf(_a + b, c)/_rf(_a, c)
                    elif c < 0:
                        return _rf(_a, b)*_rf(_a + c, -c)/_rf(_a + b + c, -c)