""" Adaptive numerical evaluation of SymPy expressions, using mpmath for mathematical functions. """ from __future__ import annotations from typing import Tuple as tTuple, Optional, Union as tUnion, Callable, List, Dict as tDict, Type, TYPE_CHECKING, \ Any, overload import math import mpmath.libmp as libmp from mpmath import ( make_mpc, make_mpf, mp, mpc, mpf, nsum, quadts, quadosc, workprec) from mpmath import inf as mpmath_inf from mpmath.libmp import (from_int, from_man_exp, from_rational, fhalf, fnan, finf, fninf, fnone, fone, fzero, mpf_abs, mpf_add, mpf_atan, mpf_atan2, mpf_cmp, mpf_cos, mpf_e, mpf_exp, mpf_log, mpf_lt, mpf_mul, mpf_neg, mpf_pi, mpf_pow, mpf_pow_int, mpf_shift, mpf_sin, mpf_sqrt, normalize, round_nearest, to_int, to_str) from mpmath.libmp import bitcount as mpmath_bitcount from mpmath.libmp.backend import MPZ from mpmath.libmp.libmpc import _infs_nan from mpmath.libmp.libmpf import dps_to_prec, prec_to_dps from .sympify import sympify from .singleton import S from sympy.external.gmpy import SYMPY_INTS from sympy.utilities.iterables import is_sequence from sympy.utilities.lambdify import lambdify from sympy.utilities.misc import as_int if TYPE_CHECKING: from sympy.core.expr import Expr from sympy.core.add import Add from sympy.core.mul import Mul from sympy.core.power import Pow from sympy.core.symbol import Symbol from sympy.integrals.integrals import Integral from sympy.concrete.summations import Sum from sympy.concrete.products import Product from sympy.functions.elementary.exponential import exp, log from sympy.functions.elementary.complexes import Abs, re, im from sympy.functions.elementary.integers import ceiling, floor from sympy.functions.elementary.trigonometric import atan from .numbers import Float, Rational, Integer, AlgebraicNumber, Number LG10 = math.log(10, 2) rnd = round_nearest def bitcount(n): """Return smallest integer, b, such that |n|/2**b < 1. """ return mpmath_bitcount(abs(int(n))) # Used in a few places as placeholder values to denote exponents and # precision levels, e.g. of exact numbers. Must be careful to avoid # passing these to mpmath functions or returning them in final results. INF = float(mpmath_inf) MINUS_INF = float(-mpmath_inf) # ~= 100 digits. Real men set this to INF. DEFAULT_MAXPREC = 333 class PrecisionExhausted(ArithmeticError): pass #----------------------------------------------------------------------------# # # # Helper functions for arithmetic and complex parts # # # #----------------------------------------------------------------------------# """ An mpf value tuple is a tuple of integers (sign, man, exp, bc) representing a floating-point number: [1, -1][sign]*man*2**exp where sign is 0 or 1 and bc should correspond to the number of bits used to represent the mantissa (man) in binary notation, e.g. """ MPF_TUP = tTuple[int, int, int, int] # mpf value tuple """ Explanation =========== >>> from sympy.core.evalf import bitcount >>> sign, man, exp, bc = 0, 5, 1, 3 >>> n = [1, -1][sign]*man*2**exp >>> n, bitcount(man) (10, 3) A temporary result is a tuple (re, im, re_acc, im_acc) where re and im are nonzero mpf value tuples representing approximate numbers, or None to denote exact zeros. re_acc, im_acc are integers denoting log2(e) where e is the estimated relative accuracy of the respective complex part, but may be anything if the corresponding complex part is None. """ TMP_RES = Any # temporary result, should be some variant of # tUnion[tTuple[Optional[MPF_TUP], Optional[MPF_TUP], # Optional[int], Optional[int]], # 'ComplexInfinity'] # but mypy reports error because it doesn't know as we know # 1. re and re_acc are either both None or both MPF_TUP # 2. sometimes the result can't be zoo # type of the "options" parameter in internal evalf functions OPT_DICT = tDict[str, Any] def fastlog(x: Optional[MPF_TUP]) -> tUnion[int, Any]: """Fast approximation of log2(x) for an mpf value tuple x. Explanation =========== Calculated as exponent + width of mantissa. This is an approximation for two reasons: 1) it gives the ceil(log2(abs(x))) value and 2) it is too high by 1 in the case that x is an exact power of 2. Although this is easy to remedy by testing to see if the odd mpf mantissa is 1 (indicating that one was dealing with an exact power of 2) that would decrease the speed and is not necessary as this is only being used as an approximation for the number of bits in x. The correct return value could be written as "x[2] + (x[3] if x[1] != 1 else 0)". Since mpf tuples always have an odd mantissa, no check is done to see if the mantissa is a multiple of 2 (in which case the result would be too large by 1). Examples ======== >>> from sympy import log >>> from sympy.core.evalf import fastlog, bitcount >>> s, m, e = 0, 5, 1 >>> bc = bitcount(m) >>> n = [1, -1][s]*m*2**e >>> n, (log(n)/log(2)).evalf(2), fastlog((s, m, e, bc)) (10, 3.3, 4) """ if not x or x == fzero: return MINUS_INF return x[2] + x[3] def pure_complex(v: 'Expr', or_real=False) -> tuple['Number', 'Number'] | None: """Return a and b if v matches a + I*b where b is not zero and a and b are Numbers, else None. If `or_real` is True then 0 will be returned for `b` if `v` is a real number. Examples ======== >>> from sympy.core.evalf import pure_complex >>> from sympy import sqrt, I, S >>> a, b, surd = S(2), S(3), sqrt(2) >>> pure_complex(a) >>> pure_complex(a, or_real=True) (2, 0) >>> pure_complex(surd) >>> pure_complex(a + b*I) (2, 3) >>> pure_complex(I) (0, 1) """ h, t = v.as_coeff_Add() if t: c, i = t.as_coeff_Mul() if i is S.ImaginaryUnit: return h, c elif or_real: return h, S.Zero return None # I don't know what this is, see function scaled_zero below SCALED_ZERO_TUP = tTuple[List[int], int, int, int] @overload def scaled_zero(mag: SCALED_ZERO_TUP, sign=1) -> MPF_TUP: ... @overload def scaled_zero(mag: int, sign=1) -> tTuple[SCALED_ZERO_TUP, int]: ... def scaled_zero(mag: tUnion[SCALED_ZERO_TUP, int], sign=1) -> \ tUnion[MPF_TUP, tTuple[SCALED_ZERO_TUP, int]]: """Return an mpf representing a power of two with magnitude ``mag`` and -1 for precision. Or, if ``mag`` is a scaled_zero tuple, then just remove the sign from within the list that it was initially wrapped in. Examples ======== >>> from sympy.core.evalf import scaled_zero >>> from sympy import Float >>> z, p = scaled_zero(100) >>> z, p (([0], 1, 100, 1), -1) >>> ok = scaled_zero(z) >>> ok (0, 1, 100, 1) >>> Float(ok) 1.26765060022823e+30 >>> Float(ok, p) 0.e+30 >>> ok, p = scaled_zero(100, -1) >>> Float(scaled_zero(ok), p) -0.e+30 """ if isinstance(mag, tuple) and len(mag) == 4 and iszero(mag, scaled=True): return (mag[0][0],) + mag[1:] elif isinstance(mag, SYMPY_INTS): if sign not in [-1, 1]: raise ValueError('sign must be +/-1') rv, p = mpf_shift(fone, mag), -1 s = 0 if sign == 1 else 1 rv = ([s],) + rv[1:] return rv, p else: raise ValueError('scaled zero expects int or scaled_zero tuple.') def iszero(mpf: tUnion[MPF_TUP, SCALED_ZERO_TUP, None], scaled=False) -> Optional[bool]: if not scaled: return not mpf or not mpf[1] and not mpf[-1] return mpf and isinstance(mpf[0], list) and mpf[1] == mpf[-1] == 1 def complex_accuracy(result: TMP_RES) -> tUnion[int, Any]: """ Returns relative accuracy of a complex number with given accuracies for the real and imaginary parts. The relative accuracy is defined in the complex norm sense as ||z|+|error|| / |z| where error is equal to (real absolute error) + (imag absolute error)*i. The full expression for the (logarithmic) error can be approximated easily by using the max norm to approximate the complex norm. In the worst case (re and im equal), this is wrong by a factor sqrt(2), or by log2(sqrt(2)) = 0.5 bit. """ if result is S.ComplexInfinity: return INF re, im, re_acc, im_acc = result if not im: if not re: return INF return re_acc if not re: return im_acc re_size = fastlog(re) im_size = fastlog(im) absolute_error = max(re_size - re_acc, im_size - im_acc) relative_error = absolute_error - max(re_size, im_size) return -relative_error def get_abs(expr: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES: result = evalf(expr, prec + 2, options) if result is S.ComplexInfinity: return finf, None, prec, None re, im, re_acc, im_acc = result if not re: re, re_acc, im, im_acc = im, im_acc, re, re_acc if im: if expr.is_number: abs_expr, _, acc, _ = evalf(abs(N(expr, prec + 2)), prec + 2, options) return abs_expr, None, acc, None else: if 'subs' in options: return libmp.mpc_abs((re, im), prec), None, re_acc, None return abs(expr), None, prec, None elif re: return mpf_abs(re), None, re_acc, None else: return None, None, None, None def get_complex_part(expr: 'Expr', no: int, prec: int, options: OPT_DICT) -> TMP_RES: """no = 0 for real part, no = 1 for imaginary part""" workprec = prec i = 0 while 1: res = evalf(expr, workprec, options) if res is S.ComplexInfinity: return fnan, None, prec, None value, accuracy = res[no::2] # XXX is the last one correct? Consider re((1+I)**2).n() if (not value) or accuracy >= prec or -value[2] > prec: return value, None, accuracy, None workprec += max(30, 2**i) i += 1 def evalf_abs(expr: 'Abs', prec: int, options: OPT_DICT) -> TMP_RES: return get_abs(expr.args[0], prec, options) def evalf_re(expr: 're', prec: int, options: OPT_DICT) -> TMP_RES: return get_complex_part(expr.args[0], 0, prec, options) def evalf_im(expr: 'im', prec: int, options: OPT_DICT) -> TMP_RES: return get_complex_part(expr.args[0], 1, prec, options) def finalize_complex(re: MPF_TUP, im: MPF_TUP, prec: int) -> TMP_RES: if re == fzero and im == fzero: raise ValueError("got complex zero with unknown accuracy") elif re == fzero: return None, im, None, prec elif im == fzero: return re, None, prec, None size_re = fastlog(re) size_im = fastlog(im) if size_re > size_im: re_acc = prec im_acc = prec + min(-(size_re - size_im), 0) else: im_acc = prec re_acc = prec + min(-(size_im - size_re), 0) return re, im, re_acc, im_acc def chop_parts(value: TMP_RES, prec: int) -> TMP_RES: """ Chop off tiny real or complex parts. """ if value is S.ComplexInfinity: return value re, im, re_acc, im_acc = value # Method 1: chop based on absolute value if re and re not in _infs_nan and (fastlog(re) < -prec + 4): re, re_acc = None, None if im and im not in _infs_nan and (fastlog(im) < -prec + 4): im, im_acc = None, None # Method 2: chop if inaccurate and relatively small if re and im: delta = fastlog(re) - fastlog(im) if re_acc < 2 and (delta - re_acc <= -prec + 4): re, re_acc = None, None if im_acc < 2 and (delta - im_acc >= prec - 4): im, im_acc = None, None return re, im, re_acc, im_acc def check_target(expr: 'Expr', result: TMP_RES, prec: int): a = complex_accuracy(result) if a < prec: raise PrecisionExhausted("Failed to distinguish the expression: \n\n%s\n\n" "from zero. Try simplifying the input, using chop=True, or providing " "a higher maxn for evalf" % (expr)) def get_integer_part(expr: 'Expr', no: int, options: OPT_DICT, return_ints=False) -> \ tUnion[TMP_RES, tTuple[int, int]]: """ With no = 1, computes ceiling(expr) With no = -1, computes floor(expr) Note: this function either gives the exact result or signals failure. """ from sympy.functions.elementary.complexes import re, im # The expression is likely less than 2^30 or so assumed_size = 30 result = evalf(expr, assumed_size, options) if result is S.ComplexInfinity: raise ValueError("Cannot get integer part of Complex Infinity") ire, iim, ire_acc, iim_acc = result # We now know the size, so we can calculate how much extra precision # (if any) is needed to get within the nearest integer if ire and iim: gap = max(fastlog(ire) - ire_acc, fastlog(iim) - iim_acc) elif ire: gap = fastlog(ire) - ire_acc elif iim: gap = fastlog(iim) - iim_acc else: # ... or maybe the expression was exactly zero if return_ints: return 0, 0 else: return None, None, None, None margin = 10 if gap >= -margin: prec = margin + assumed_size + gap ire, iim, ire_acc, iim_acc = evalf( expr, prec, options) else: prec = assumed_size # We can now easily find the nearest integer, but to find floor/ceil, we # must also calculate whether the difference to the nearest integer is # positive or negative (which may fail if very close). def calc_part(re_im: 'Expr', nexpr: MPF_TUP): from .add import Add _, _, exponent, _ = nexpr is_int = exponent == 0 nint = int(to_int(nexpr, rnd)) if is_int: # make sure that we had enough precision to distinguish # between nint and the re or im part (re_im) of expr that # was passed to calc_part ire, iim, ire_acc, iim_acc = evalf( re_im - nint, 10, options) # don't need much precision assert not iim size = -fastlog(ire) + 2 # -ve b/c ire is less than 1 if size > prec: ire, iim, ire_acc, iim_acc = evalf( re_im, size, options) assert not iim nexpr = ire nint = int(to_int(nexpr, rnd)) _, _, new_exp, _ = ire is_int = new_exp == 0 if not is_int: # if there are subs and they all contain integer re/im parts # then we can (hopefully) safely substitute them into the # expression s = options.get('subs', False) if s: doit = True # use strict=False with as_int because we take # 2.0 == 2 for v in s.values(): try: as_int(v, strict=False) except ValueError: try: [as_int(i, strict=False) for i in v.as_real_imag()] continue except (ValueError, AttributeError): doit = False break if doit: re_im = re_im.subs(s) re_im = Add(re_im, -nint, evaluate=False) x, _, x_acc, _ = evalf(re_im, 10, options) try: check_target(re_im, (x, None, x_acc, None), 3) except PrecisionExhausted: if not re_im.equals(0): raise PrecisionExhausted x = fzero nint += int(no*(mpf_cmp(x or fzero, fzero) == no)) nint = from_int(nint) return nint, INF re_, im_, re_acc, im_acc = None, None, None, None if ire: re_, re_acc = calc_part(re(expr, evaluate=False), ire) if iim: im_, im_acc = calc_part(im(expr, evaluate=False), iim) if return_ints: return int(to_int(re_ or fzero)), int(to_int(im_ or fzero)) return re_, im_, re_acc, im_acc def evalf_ceiling(expr: 'ceiling', prec: int, options: OPT_DICT) -> TMP_RES: return get_integer_part(expr.args[0], 1, options) def evalf_floor(expr: 'floor', prec: int, options: OPT_DICT) -> TMP_RES: return get_integer_part(expr.args[0], -1, options) def evalf_float(expr: 'Float', prec: int, options: OPT_DICT) -> TMP_RES: return expr._mpf_, None, prec, None def evalf_rational(expr: 'Rational', prec: int, options: OPT_DICT) -> TMP_RES: return from_rational(expr.p, expr.q, prec), None, prec, None def evalf_integer(expr: 'Integer', prec: int, options: OPT_DICT) -> TMP_RES: return from_int(expr.p, prec), None, prec, None #----------------------------------------------------------------------------# # # # Arithmetic operations # # # #----------------------------------------------------------------------------# def add_terms(terms: list, prec: int, target_prec: int) -> \ tTuple[tUnion[MPF_TUP, SCALED_ZERO_TUP, None], Optional[int]]: """ Helper for evalf_add. Adds a list of (mpfval, accuracy) terms. Returns ======= - None, None if there are no non-zero terms; - terms[0] if there is only 1 term; - scaled_zero if the sum of the terms produces a zero by cancellation e.g. mpfs representing 1 and -1 would produce a scaled zero which need special handling since they are not actually zero and they are purposely malformed to ensure that they cannot be used in anything but accuracy calculations; - a tuple that is scaled to target_prec that corresponds to the sum of the terms. The returned mpf tuple will be normalized to target_prec; the input prec is used to define the working precision. XXX explain why this is needed and why one cannot just loop using mpf_add """ terms = [t for t in terms if not iszero(t[0])] if not terms: return None, None elif len(terms) == 1: return terms[0] # see if any argument is NaN or oo and thus warrants a special return special = [] from .numbers import Float for t in terms: arg = Float._new(t[0], 1) if arg is S.NaN or arg.is_infinite: special.append(arg) if special: from .add import Add rv = evalf(Add(*special), prec + 4, {}) return rv[0], rv[2] working_prec = 2*prec sum_man, sum_exp = 0, 0 absolute_err: List[int] = [] for x, accuracy in terms: sign, man, exp, bc = x if sign: man = -man absolute_err.append(bc + exp - accuracy) delta = exp - sum_exp if exp >= sum_exp: # x much larger than existing sum? # first: quick test if ((delta > working_prec) and ((not sum_man) or delta - bitcount(abs(sum_man)) > working_prec)): sum_man = man sum_exp = exp else: sum_man += (man << delta) else: delta = -delta # x much smaller than existing sum? if delta - bc > working_prec: if not sum_man: sum_man, sum_exp = man, exp else: sum_man = (sum_man << delta) + man sum_exp = exp absolute_error = max(absolute_err) if not sum_man: return scaled_zero(absolute_error) if sum_man < 0: sum_sign = 1 sum_man = -sum_man else: sum_sign = 0 sum_bc = bitcount(sum_man) sum_accuracy = sum_exp + sum_bc - absolute_error r = normalize(sum_sign, sum_man, sum_exp, sum_bc, target_prec, rnd), sum_accuracy return r def evalf_add(v: 'Add', prec: int, options: OPT_DICT) -> TMP_RES: res = pure_complex(v) if res: h, c = res re, _, re_acc, _ = evalf(h, prec, options) im, _, im_acc, _ = evalf(c, prec, options) return re, im, re_acc, im_acc oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC) i = 0 target_prec = prec while 1: options['maxprec'] = min(oldmaxprec, 2*prec) terms = [evalf(arg, prec + 10, options) for arg in v.args] n = terms.count(S.ComplexInfinity) if n >= 2: return fnan, None, prec, None re, re_acc = add_terms( [a[0::2] for a in terms if isinstance(a, tuple) and a[0]], prec, target_prec) im, im_acc = add_terms( [a[1::2] for a in terms if isinstance(a, tuple) and a[1]], prec, target_prec) if n == 1: if re in (finf, fninf, fnan) or im in (finf, fninf, fnan): return fnan, None, prec, None return S.ComplexInfinity acc = complex_accuracy((re, im, re_acc, im_acc)) if acc >= target_prec: if options.get('verbose'): print("ADD: wanted", target_prec, "accurate bits, got", re_acc, im_acc) break else: if (prec - target_prec) > options['maxprec']: break prec = prec + max(10 + 2**i, target_prec - acc) i += 1 if options.get('verbose'): print("ADD: restarting with prec", prec) options['maxprec'] = oldmaxprec if iszero(re, scaled=True): re = scaled_zero(re) if iszero(im, scaled=True): im = scaled_zero(im) return re, im, re_acc, im_acc def evalf_mul(v: 'Mul', prec: int, options: OPT_DICT) -> TMP_RES: res = pure_complex(v) if res: # the only pure complex that is a mul is h*I _, h = res im, _, im_acc, _ = evalf(h, prec, options) return None, im, None, im_acc args = list(v.args) # see if any argument is NaN or oo and thus warrants a special return has_zero = False special = [] from .numbers import Float for arg in args: result = evalf(arg, prec, options) if result is S.ComplexInfinity: special.append(result) continue if result[0] is None: if result[1] is None: has_zero = True continue num = Float._new(result[0], 1) if num is S.NaN: return fnan, None, prec, None if num.is_infinite: special.append(num) if special: if has_zero: return fnan, None, prec, None from .mul import Mul return evalf(Mul(*special), prec + 4, {}) if has_zero: return None, None, None, None # With guard digits, multiplication in the real case does not destroy # accuracy. This is also true in the complex case when considering the # total accuracy; however accuracy for the real or imaginary parts # separately may be lower. acc = prec # XXX: big overestimate working_prec = prec + len(args) + 5 # Empty product is 1 start = man, exp, bc = MPZ(1), 0, 1 # First, we multiply all pure real or pure imaginary numbers. # direction tells us that the result should be multiplied by # I**direction; all other numbers get put into complex_factors # to be multiplied out after the first phase. last = len(args) direction = 0 args.append(S.One) complex_factors = [] for i, arg in enumerate(args): if i != last and pure_complex(arg): args[-1] = (args[-1]*arg).expand() continue elif i == last and arg is S.One: continue re, im, re_acc, im_acc = evalf(arg, working_prec, options) if re and im: complex_factors.append((re, im, re_acc, im_acc)) continue elif re: (s, m, e, b), w_acc = re, re_acc elif im: (s, m, e, b), w_acc = im, im_acc direction += 1 else: return None, None, None, None direction += 2*s man *= m exp += e bc += b while bc > 3*working_prec: man >>= working_prec exp += working_prec bc -= working_prec acc = min(acc, w_acc) sign = (direction & 2) >> 1 if not complex_factors: v = normalize(sign, man, exp, bitcount(man), prec, rnd) # multiply by i if direction & 1: return None, v, None, acc else: return v, None, acc, None else: # initialize with the first term if (man, exp, bc) != start: # there was a real part; give it an imaginary part re, im = (sign, man, exp, bitcount(man)), (0, MPZ(0), 0, 0) i0 = 0 else: # there is no real part to start (other than the starting 1) wre, wim, wre_acc, wim_acc = complex_factors[0] acc = min(acc, complex_accuracy((wre, wim, wre_acc, wim_acc))) re = wre im = wim i0 = 1 for wre, wim, wre_acc, wim_acc in complex_factors[i0:]: # acc is the overall accuracy of the product; we aren't # computing exact accuracies of the product. acc = min(acc, complex_accuracy((wre, wim, wre_acc, wim_acc))) use_prec = working_prec A = mpf_mul(re, wre, use_prec) B = mpf_mul(mpf_neg(im), wim, use_prec) C = mpf_mul(re, wim, use_prec) D = mpf_mul(im, wre, use_prec) re = mpf_add(A, B, use_prec) im = mpf_add(C, D, use_prec) if options.get('verbose'): print("MUL: wanted", prec, "accurate bits, got", acc) # multiply by I if direction & 1: re, im = mpf_neg(im), re return re, im, acc, acc def evalf_pow(v: 'Pow', prec: int, options) -> TMP_RES: target_prec = prec base, exp = v.args # We handle x**n separately. This has two purposes: 1) it is much # faster, because we avoid calling evalf on the exponent, and 2) it # allows better handling of real/imaginary parts that are exactly zero if exp.is_Integer: p: int = exp.p # type: ignore # Exact if not p: return fone, None, prec, None # Exponentiation by p magnifies relative error by |p|, so the # base must be evaluated with increased precision if p is large prec += int(math.log(abs(p), 2)) result = evalf(base, prec + 5, options) if result is S.ComplexInfinity: if p < 0: return None, None, None, None return result re, im, re_acc, im_acc = result # Real to integer power if re and not im: return mpf_pow_int(re, p, target_prec), None, target_prec, None # (x*I)**n = I**n * x**n if im and not re: z = mpf_pow_int(im, p, target_prec) case = p % 4 if case == 0: return z, None, target_prec, None if case == 1: return None, z, None, target_prec if case == 2: return mpf_neg(z), None, target_prec, None if case == 3: return None, mpf_neg(z), None, target_prec # Zero raised to an integer power if not re: if p < 0: return S.ComplexInfinity return None, None, None, None # General complex number to arbitrary integer power re, im = libmp.mpc_pow_int((re, im), p, prec) # Assumes full accuracy in input return finalize_complex(re, im, target_prec) result = evalf(base, prec + 5, options) if result is S.ComplexInfinity: if exp.is_Rational: if exp < 0: return None, None, None, None return result raise NotImplementedError # Pure square root if exp is S.Half: xre, xim, _, _ = result # General complex square root if xim: re, im = libmp.mpc_sqrt((xre or fzero, xim), prec) return finalize_complex(re, im, prec) if not xre: return None, None, None, None # Square root of a negative real number if mpf_lt(xre, fzero): return None, mpf_sqrt(mpf_neg(xre), prec), None, prec # Positive square root return mpf_sqrt(xre, prec), None, prec, None # We first evaluate the exponent to find its magnitude # This determines the working precision that must be used prec += 10 result = evalf(exp, prec, options) if result is S.ComplexInfinity: return fnan, None, prec, None yre, yim, _, _ = result # Special cases: x**0 if not (yre or yim): return fone, None, prec, None ysize = fastlog(yre) # Restart if too big # XXX: prec + ysize might exceed maxprec if ysize > 5: prec += ysize yre, yim, _, _ = evalf(exp, prec, options) # Pure exponential function; no need to evalf the base if base is S.Exp1: if yim: re, im = libmp.mpc_exp((yre or fzero, yim), prec) return finalize_complex(re, im, target_prec) return mpf_exp(yre, target_prec), None, target_prec, None xre, xim, _, _ = evalf(base, prec + 5, options) # 0**y if not (xre or xim): if yim: return fnan, None, prec, None if yre[0] == 1: # y < 0 return S.ComplexInfinity return None, None, None, None # (real ** complex) or (complex ** complex) if yim: re, im = libmp.mpc_pow( (xre or fzero, xim or fzero), (yre or fzero, yim), target_prec) return finalize_complex(re, im, target_prec) # complex ** real if xim: re, im = libmp.mpc_pow_mpf((xre or fzero, xim), yre, target_prec) return finalize_complex(re, im, target_prec) # negative ** real elif mpf_lt(xre, fzero): re, im = libmp.mpc_pow_mpf((xre, fzero), yre, target_prec) return finalize_complex(re, im, target_prec) # positive ** real else: return mpf_pow(xre, yre, target_prec), None, target_prec, None #----------------------------------------------------------------------------# # # # Special functions # # # #----------------------------------------------------------------------------# def evalf_exp(expr: 'exp', prec: int, options: OPT_DICT) -> TMP_RES: from .power import Pow return evalf_pow(Pow(S.Exp1, expr.exp, evaluate=False), prec, options) def evalf_trig(v: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES: """ This function handles sin and cos of complex arguments. TODO: should also handle tan of complex arguments. """ from sympy.functions.elementary.trigonometric import cos, sin if isinstance(v, cos): func = mpf_cos elif isinstance(v, sin): func = mpf_sin else: raise NotImplementedError arg = v.args[0] # 20 extra bits is possibly overkill. It does make the need # to restart very unlikely xprec = prec + 20 re, im, re_acc, im_acc = evalf(arg, xprec, options) if im: if 'subs' in options: v = v.subs(options['subs']) return evalf(v._eval_evalf(prec), prec, options) if not re: if isinstance(v, cos): return fone, None, prec, None elif isinstance(v, sin): return None, None, None, None else: raise NotImplementedError # For trigonometric functions, we are interested in the # fixed-point (absolute) accuracy of the argument. xsize = fastlog(re) # Magnitude <= 1.0. OK to compute directly, because there is no # danger of hitting the first root of cos (with sin, magnitude # <= 2.0 would actually be ok) if xsize < 1: return func(re, prec, rnd), None, prec, None # Very large if xsize >= 10: xprec = prec + xsize re, im, re_acc, im_acc = evalf(arg, xprec, options) # Need to repeat in case the argument is very close to a # multiple of pi (or pi/2), hitting close to a root while 1: y = func(re, prec, rnd) ysize = fastlog(y) gap = -ysize accuracy = (xprec - xsize) - gap if accuracy < prec: if options.get('verbose'): print("SIN/COS", accuracy, "wanted", prec, "gap", gap) print(to_str(y, 10)) if xprec > options.get('maxprec', DEFAULT_MAXPREC): return y, None, accuracy, None xprec += gap re, im, re_acc, im_acc = evalf(arg, xprec, options) continue else: return y, None, prec, None def evalf_log(expr: 'log', prec: int, options: OPT_DICT) -> TMP_RES: if len(expr.args)>1: expr = expr.doit() return evalf(expr, prec, options) arg = expr.args[0] workprec = prec + 10 result = evalf(arg, workprec, options) if result is S.ComplexInfinity: return result xre, xim, xacc, _ = result # evalf can return NoneTypes if chop=True # issue 18516, 19623 if xre is xim is None: # Dear reviewer, I do not know what -inf is; # it looks to be (1, 0, -789, -3) # but I'm not sure in general, # so we just let mpmath figure # it out by taking log of 0 directly. # It would be better to return -inf instead. xre = fzero if xim: from sympy.functions.elementary.complexes import Abs from sympy.functions.elementary.exponential import log # XXX: use get_abs etc instead re = evalf_log( log(Abs(arg, evaluate=False), evaluate=False), prec, options) im = mpf_atan2(xim, xre or fzero, prec) return re[0], im, re[2], prec imaginary_term = (mpf_cmp(xre, fzero) < 0) re = mpf_log(mpf_abs(xre), prec, rnd) size = fastlog(re) if prec - size > workprec and re != fzero: from .add import Add # We actually need to compute 1+x accurately, not x add = Add(S.NegativeOne, arg, evaluate=False) xre, xim, _, _ = evalf_add(add, prec, options) prec2 = workprec - fastlog(xre) # xre is now x - 1 so we add 1 back here to calculate x re = mpf_log(mpf_abs(mpf_add(xre, fone, prec2)), prec, rnd) re_acc = prec if imaginary_term: return re, mpf_pi(prec), re_acc, prec else: return re, None, re_acc, None def evalf_atan(v: 'atan', prec: int, options: OPT_DICT) -> TMP_RES: arg = v.args[0] xre, xim, reacc, imacc = evalf(arg, prec + 5, options) if xre is xim is None: return (None,)*4 if xim: raise NotImplementedError return mpf_atan(xre, prec, rnd), None, prec, None def evalf_subs(prec: int, subs: dict) -> dict: """ Change all Float entries in `subs` to have precision prec. """ newsubs = {} for a, b in subs.items(): b = S(b) if b.is_Float: b = b._eval_evalf(prec) newsubs[a] = b return newsubs def evalf_piecewise(expr: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES: from .numbers import Float, Integer if 'subs' in options: expr = expr.subs(evalf_subs(prec, options['subs'])) newopts = options.copy() del newopts['subs'] if hasattr(expr, 'func'): return evalf(expr, prec, newopts) if isinstance(expr, float): return evalf(Float(expr), prec, newopts) if isinstance(expr, int): return evalf(Integer(expr), prec, newopts) # We still have undefined symbols raise NotImplementedError def evalf_alg_num(a: 'AlgebraicNumber', prec: int, options: OPT_DICT) -> TMP_RES: return evalf(a.to_root(), prec, options) #----------------------------------------------------------------------------# # # # High-level operations # # # #----------------------------------------------------------------------------# def as_mpmath(x: Any, prec: int, options: OPT_DICT) -> tUnion[mpc, mpf]: from .numbers import Infinity, NegativeInfinity, Zero x = sympify(x) if isinstance(x, Zero) or x == 0.0: return mpf(0) if isinstance(x, Infinity): return mpf('inf') if isinstance(x, NegativeInfinity): return mpf('-inf') # XXX result = evalf(x, prec, options) return quad_to_mpmath(result) def do_integral(expr: 'Integral', prec: int, options: OPT_DICT) -> TMP_RES: func = expr.args[0] x, xlow, xhigh = expr.args[1] if xlow == xhigh: xlow = xhigh = 0 elif x not in func.free_symbols: # only the difference in limits matters in this case # so if there is a symbol in common that will cancel # out when taking the difference, then use that # difference if xhigh.free_symbols & xlow.free_symbols: diff = xhigh - xlow if diff.is_number: xlow, xhigh = 0, diff oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC) options['maxprec'] = min(oldmaxprec, 2*prec) with workprec(prec + 5): xlow = as_mpmath(xlow, prec + 15, options) xhigh = as_mpmath(xhigh, prec + 15, options) # Integration is like summation, and we can phone home from # the integrand function to update accuracy summation style # Note that this accuracy is inaccurate, since it fails # to account for the variable quadrature weights, # but it is better than nothing from sympy.functions.elementary.trigonometric import cos, sin from .symbol import Wild have_part = [False, False] max_real_term: tUnion[float, int] = MINUS_INF max_imag_term: tUnion[float, int] = MINUS_INF def f(t: 'Expr') -> tUnion[mpc, mpf]: nonlocal max_real_term, max_imag_term re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}}) have_part[0] = re or have_part[0] have_part[1] = im or have_part[1] max_real_term = max(max_real_term, fastlog(re)) max_imag_term = max(max_imag_term, fastlog(im)) if im: return mpc(re or fzero, im) return mpf(re or fzero) if options.get('quad') == 'osc': A = Wild('A', exclude=[x]) B = Wild('B', exclude=[x]) D = Wild('D') m = func.match(cos(A*x + B)*D) if not m: m = func.match(sin(A*x + B)*D) if not m: raise ValueError("An integrand of the form sin(A*x+B)*f(x) " "or cos(A*x+B)*f(x) is required for oscillatory quadrature") period = as_mpmath(2*S.Pi/m[A], prec + 15, options) result = quadosc(f, [xlow, xhigh], period=period) # XXX: quadosc does not do error detection yet quadrature_error = MINUS_INF else: result, quadrature_err = quadts(f, [xlow, xhigh], error=1) quadrature_error = fastlog(quadrature_err._mpf_) options['maxprec'] = oldmaxprec if have_part[0]: re: Optional[MPF_TUP] = result.real._mpf_ re_acc: Optional[int] if re == fzero: re_s, re_acc = scaled_zero(int(-max(prec, max_real_term, quadrature_error))) re = scaled_zero(re_s) # handled ok in evalf_integral else: re_acc = int(-max(max_real_term - fastlog(re) - prec, quadrature_error)) else: re, re_acc = None, None if have_part[1]: im: Optional[MPF_TUP] = result.imag._mpf_ im_acc: Optional[int] if im == fzero: im_s, im_acc = scaled_zero(int(-max(prec, max_imag_term, quadrature_error))) im = scaled_zero(im_s) # handled ok in evalf_integral else: im_acc = int(-max(max_imag_term - fastlog(im) - prec, quadrature_error)) else: im, im_acc = None, None result = re, im, re_acc, im_acc return result def evalf_integral(expr: 'Integral', prec: int, options: OPT_DICT) -> TMP_RES: limits = expr.limits if len(limits) != 1 or len(limits[0]) != 3: raise NotImplementedError workprec = prec i = 0 maxprec = options.get('maxprec', INF) while 1: result = do_integral(expr, workprec, options) accuracy = complex_accuracy(result) if accuracy >= prec: # achieved desired precision break if workprec >= maxprec: # can't increase accuracy any more break if accuracy == -1: # maybe the answer really is zero and maybe we just haven't increased # the precision enough. So increase by doubling to not take too long # to get to maxprec. workprec *= 2 else: workprec += max(prec, 2**i) workprec = min(workprec, maxprec) i += 1 return result def check_convergence(numer: 'Expr', denom: 'Expr', n: 'Symbol') -> tTuple[int, Any, Any]: """ Returns ======= (h, g, p) where -- h is: > 0 for convergence of rate 1/factorial(n)**h < 0 for divergence of rate factorial(n)**(-h) = 0 for geometric or polynomial convergence or divergence -- abs(g) is: > 1 for geometric convergence of rate 1/h**n < 1 for geometric divergence of rate h**n = 1 for polynomial convergence or divergence (g < 0 indicates an alternating series) -- p is: > 1 for polynomial convergence of rate 1/n**h <= 1 for polynomial divergence of rate n**(-h) """ from sympy.polys.polytools import Poly npol = Poly(numer, n) dpol = Poly(denom, n) p = npol.degree() q = dpol.degree() rate = q - p if rate: return rate, None, None constant = dpol.LC() / npol.LC() from .numbers import equal_valued if not equal_valued(abs(constant), 1): return rate, constant, None if npol.degree() == dpol.degree() == 0: return rate, constant, 0 pc = npol.all_coeffs()[1] qc = dpol.all_coeffs()[1] return rate, constant, (qc - pc)/dpol.LC() def hypsum(expr: 'Expr', n: 'Symbol', start: int, prec: int) -> mpf: """ Sum a rapidly convergent infinite hypergeometric series with given general term, e.g. e = hypsum(1/factorial(n), n). The quotient between successive terms must be a quotient of integer polynomials. """ from .numbers import Float, equal_valued from sympy.simplify.simplify import hypersimp if prec == float('inf'): raise NotImplementedError('does not support inf prec') if start: expr = expr.subs(n, n + start) hs = hypersimp(expr, n) if hs is None: raise NotImplementedError("a hypergeometric series is required") num, den = hs.as_numer_denom() func1 = lambdify(n, num) func2 = lambdify(n, den) h, g, p = check_convergence(num, den, n) if h < 0: raise ValueError("Sum diverges like (n!)^%i" % (-h)) term = expr.subs(n, 0) if not term.is_Rational: raise NotImplementedError("Non rational term functionality is not implemented.") # Direct summation if geometric or faster if h > 0 or (h == 0 and abs(g) > 1): term = (MPZ(term.p) << prec) // term.q s = term k = 1 while abs(term) > 5: term *= MPZ(func1(k - 1)) term //= MPZ(func2(k - 1)) s += term k += 1 return from_man_exp(s, -prec) else: alt = g < 0 if abs(g) < 1: raise ValueError("Sum diverges like (%i)^n" % abs(1/g)) if p < 1 or (equal_valued(p, 1) and not alt): raise ValueError("Sum diverges like n^%i" % (-p)) # We have polynomial convergence: use Richardson extrapolation vold = None ndig = prec_to_dps(prec) while True: # Need to use at least quad precision because a lot of cancellation # might occur in the extrapolation process; we check the answer to # make sure that the desired precision has been reached, too. prec2 = 4*prec term0 = (MPZ(term.p) << prec2) // term.q def summand(k, _term=[term0]): if k: k = int(k) _term[0] *= MPZ(func1(k - 1)) _term[0] //= MPZ(func2(k - 1)) return make_mpf(from_man_exp(_term[0], -prec2)) with workprec(prec): v = nsum(summand, [0, mpmath_inf], method='richardson') vf = Float(v, ndig) if vold is not None and vold == vf: break prec += prec # double precision each time vold = vf return v._mpf_ def evalf_prod(expr: 'Product', prec: int, options: OPT_DICT) -> TMP_RES: if all((l[1] - l[2]).is_Integer for l in expr.limits): result = evalf(expr.doit(), prec=prec, options=options) else: from sympy.concrete.summations import Sum result = evalf(expr.rewrite(Sum), prec=prec, options=options) return result def evalf_sum(expr: 'Sum', prec: int, options: OPT_DICT) -> TMP_RES: from .numbers import Float if 'subs' in options: expr = expr.subs(options['subs']) func = expr.function limits = expr.limits if len(limits) != 1 or len(limits[0]) != 3: raise NotImplementedError if func.is_zero: return None, None, prec, None prec2 = prec + 10 try: n, a, b = limits[0] if b is not S.Infinity or a is S.NegativeInfinity or a != int(a): raise NotImplementedError # Use fast hypergeometric summation if possible v = hypsum(func, n, int(a), prec2) delta = prec - fastlog(v) if fastlog(v) < -10: v = hypsum(func, n, int(a), delta) return v, None, min(prec, delta), None except NotImplementedError: # Euler-Maclaurin summation for general series eps = Float(2.0)**(-prec) for i in range(1, 5): m = n = 2**i * prec s, err = expr.euler_maclaurin(m=m, n=n, eps=eps, eval_integral=False) err = err.evalf() if err is S.NaN: raise NotImplementedError if err <= eps: break err = fastlog(evalf(abs(err), 20, options)[0]) re, im, re_acc, im_acc = evalf(s, prec2, options) if re_acc is None: re_acc = -err if im_acc is None: im_acc = -err return re, im, re_acc, im_acc #----------------------------------------------------------------------------# # # # Symbolic interface # # # #----------------------------------------------------------------------------# def evalf_symbol(x: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES: val = options['subs'][x] if isinstance(val, mpf): if not val: return None, None, None, None return val._mpf_, None, prec, None else: if '_cache' not in options: options['_cache'] = {} cache = options['_cache'] cached, cached_prec = cache.get(x, (None, MINUS_INF)) if cached_prec >= prec: return cached v = evalf(sympify(val), prec, options) cache[x] = (v, prec) return v evalf_table: tDict[Type['Expr'], Callable[['Expr', int, OPT_DICT], TMP_RES]] = {} def _create_evalf_table(): global evalf_table from sympy.concrete.products import Product from sympy.concrete.summations import Sum from .add import Add from .mul import Mul from .numbers import Exp1, Float, Half, ImaginaryUnit, Integer, NaN, NegativeOne, One, Pi, Rational, \ Zero, ComplexInfinity, AlgebraicNumber from .power import Pow from .symbol import Dummy, Symbol from sympy.functions.elementary.complexes import Abs, im, re from sympy.functions.elementary.exponential import exp, log from sympy.functions.elementary.integers import ceiling, floor from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.elementary.trigonometric import atan, cos, sin from sympy.integrals.integrals import Integral evalf_table = { Symbol: evalf_symbol, Dummy: evalf_symbol, Float: evalf_float, Rational: evalf_rational, Integer: evalf_integer, Zero: lambda x, prec, options: (None, None, prec, None), One: lambda x, prec, options: (fone, None, prec, None), Half: lambda x, prec, options: (fhalf, None, prec, None), Pi: lambda x, prec, options: (mpf_pi(prec), None, prec, None), Exp1: lambda x, prec, options: (mpf_e(prec), None, prec, None), ImaginaryUnit: lambda x, prec, options: (None, fone, None, prec), NegativeOne: lambda x, prec, options: (fnone, None, prec, None), ComplexInfinity: lambda x, prec, options: S.ComplexInfinity, NaN: lambda x, prec, options: (fnan, None, prec, None), exp: evalf_exp, cos: evalf_trig, sin: evalf_trig, Add: evalf_add, Mul: evalf_mul, Pow: evalf_pow, log: evalf_log, atan: evalf_atan, Abs: evalf_abs, re: evalf_re, im: evalf_im, floor: evalf_floor, ceiling: evalf_ceiling, Integral: evalf_integral, Sum: evalf_sum, Product: evalf_prod, Piecewise: evalf_piecewise, AlgebraicNumber: evalf_alg_num, } def evalf(x: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES: """ Evaluate the ``Expr`` instance, ``x`` to a binary precision of ``prec``. This function is supposed to be used internally. Parameters ========== x : Expr The formula to evaluate to a float. prec : int The binary precision that the output should have. options : dict A dictionary with the same entries as ``EvalfMixin.evalf`` and in addition, ``maxprec`` which is the maximum working precision. Returns ======= An optional tuple, ``(re, im, re_acc, im_acc)`` which are the real, imaginary, real accuracy and imaginary accuracy respectively. ``re`` is an mpf value tuple and so is ``im``. ``re_acc`` and ``im_acc`` are ints. NB: all these return values can be ``None``. If all values are ``None``, then that represents 0. Note that 0 is also represented as ``fzero = (0, 0, 0, 0)``. """ from sympy.functions.elementary.complexes import re as re_, im as im_ try: rf = evalf_table[type(x)] r = rf(x, prec, options) except KeyError: # Fall back to ordinary evalf if possible if 'subs' in options: x = x.subs(evalf_subs(prec, options['subs'])) xe = x._eval_evalf(prec) if xe is None: raise NotImplementedError as_real_imag = getattr(xe, "as_real_imag", None) if as_real_imag is None: raise NotImplementedError # e.g. FiniteSet(-1.0, 1.0).evalf() re, im = as_real_imag() if re.has(re_) or im.has(im_): raise NotImplementedError if re == 0.0: re = None reprec = None elif re.is_number: re = re._to_mpmath(prec, allow_ints=False)._mpf_ reprec = prec else: raise NotImplementedError if im == 0.0: im = None imprec = None elif im.is_number: im = im._to_mpmath(prec, allow_ints=False)._mpf_ imprec = prec else: raise NotImplementedError r = re, im, reprec, imprec if options.get("verbose"): print("### input", x) print("### output", to_str(r[0] or fzero, 50) if isinstance(r, tuple) else r) print("### raw", r) # r[0], r[2] print() chop = options.get('chop', False) if chop: if chop is True: chop_prec = prec else: # convert (approximately) from given tolerance; # the formula here will will make 1e-i rounds to 0 for # i in the range +/-27 while 2e-i will not be chopped chop_prec = int(round(-3.321*math.log10(chop) + 2.5)) if chop_prec == 3: chop_prec -= 1 r = chop_parts(r, chop_prec) if options.get("strict"): check_target(x, r, prec) return r def quad_to_mpmath(q, ctx=None): """Turn the quad returned by ``evalf`` into an ``mpf`` or ``mpc``. """ mpc = make_mpc if ctx is None else ctx.make_mpc mpf = make_mpf if ctx is None else ctx.make_mpf if q is S.ComplexInfinity: raise NotImplementedError re, im, _, _ = q if im: if not re: re = fzero return mpc((re, im)) elif re: return mpf(re) else: return mpf(fzero) class EvalfMixin: """Mixin class adding evalf capability.""" __slots__ = () # type: tTuple[str, ...] def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False): """ Evaluate the given formula to an accuracy of *n* digits. Parameters ========== subs : dict, optional Substitute numerical values for symbols, e.g. ``subs={x:3, y:1+pi}``. The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of "small" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise ``PrecisionExhausted`` if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try ``quad='osc'``. verbose : bool, optional Print debug information. Notes ===== When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: >>> from sympy.abc import x, y, z >>> values = {x: 1e16, y: 1, z: 1e16} >>> (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: >>> (x + y - z).evalf(subs=values) 1.00000000000000 """ from .numbers import Float, Number n = n if n is not None else 15 if subs and is_sequence(subs): raise TypeError('subs must be given as a dictionary') # for sake of sage that doesn't like evalf(1) if n == 1 and isinstance(self, Number): from .expr import _mag rv = self.evalf(2, subs, maxn, chop, strict, quad, verbose) m = _mag(rv) rv = rv.round(1 - m) return rv if not evalf_table: _create_evalf_table() prec = dps_to_prec(n) options = {'maxprec': max(prec, int(maxn*LG10)), 'chop': chop, 'strict': strict, 'verbose': verbose} if subs is not None: options['subs'] = subs if quad is not None: options['quad'] = quad try: result = evalf(self, prec + 4, options) except NotImplementedError: # Fall back to the ordinary evalf if hasattr(self, 'subs') and subs is not None: # issue 20291 v = self.subs(subs)._eval_evalf(prec) else: v = self._eval_evalf(prec) if v is None: return self elif not v.is_number: return v try: # If the result is numerical, normalize it result = evalf(v, prec, options) except NotImplementedError: # Probably contains symbols or unknown functions return v if result is S.ComplexInfinity: return result re, im, re_acc, im_acc = result if re is S.NaN or im is S.NaN: return S.NaN if re: p = max(min(prec, re_acc), 1) re = Float._new(re, p) else: re = S.Zero if im: p = max(min(prec, im_acc), 1) im = Float._new(im, p) return re + im*S.ImaginaryUnit else: return re n = evalf def _evalf(self, prec): """Helper for evalf. Does the same thing but takes binary precision""" r = self._eval_evalf(prec) if r is None: r = self return r def _eval_evalf(self, prec): return def _to_mpmath(self, prec, allow_ints=True): # mpmath functions accept ints as input errmsg = "cannot convert to mpmath number" if allow_ints and self.is_Integer: return self.p if hasattr(self, '_as_mpf_val'): return make_mpf(self._as_mpf_val(prec)) try: result = evalf(self, prec, {}) return quad_to_mpmath(result) except NotImplementedError: v = self._eval_evalf(prec) if v is None: raise ValueError(errmsg) if v.is_Float: return make_mpf(v._mpf_) # Number + Number*I is also fine re, im = v.as_real_imag() if allow_ints and re.is_Integer: re = from_int(re.p) elif re.is_Float: re = re._mpf_ else: raise ValueError(errmsg) if allow_ints and im.is_Integer: im = from_int(im.p) elif im.is_Float: im = im._mpf_ else: raise ValueError(errmsg) return make_mpc((re, im)) def N(x, n=15, **options): r""" Calls x.evalf(n, \*\*options). Explanations ============ Both .n() and N() are equivalent to .evalf(); use the one that you like better. See also the docstring of .evalf() for information on the options. Examples ======== >>> from sympy import Sum, oo, N >>> from sympy.abc import k >>> Sum(1/k**k, (k, 1, oo)) Sum(k**(-k), (k, 1, oo)) >>> N(_, 4) 1.291 """ # by using rational=True, any evaluation of a string # will be done using exact values for the Floats return sympify(x, rational=True).evalf(n, **options) def _evalf_with_bounded_error(x: 'Expr', eps: 'Optional[Expr]' = None, m: int = 0, options: Optional[OPT_DICT] = None) -> TMP_RES: """ Evaluate *x* to within a bounded absolute error. Parameters ========== x : Expr The quantity to be evaluated. eps : Expr, None, optional (default=None) Positive real upper bound on the acceptable error. m : int, optional (default=0) If *eps* is None, then use 2**(-m) as the upper bound on the error. options: OPT_DICT As in the ``evalf`` function. Returns ======= A tuple ``(re, im, re_acc, im_acc)``, as returned by ``evalf``. See Also ======== evalf """ if eps is not None: if not (eps.is_Rational or eps.is_Float) or not eps > 0: raise ValueError("eps must be positive") r, _, _, _ = evalf(1/eps, 1, {}) m = fastlog(r) c, d, _, _ = evalf(x, 1, {}) # Note: If x = a + b*I, then |a| <= 2|c| and |b| <= 2|d|, with equality # only in the zero case. # If a is non-zero, then |c| = 2**nc for some integer nc, and c has # bitcount 1. Therefore 2**fastlog(c) = 2**(nc+1) = 2|c| is an upper bound # on |a|. Likewise for b and d. nr, ni = fastlog(c), fastlog(d) n = max(nr, ni) + 1 # If x is 0, then n is MINUS_INF, and p will be 1. Otherwise, # n - 1 bits get us past the integer parts of a and b, and +1 accounts for # the factor of <= sqrt(2) that is |x|/max(|a|, |b|). p = max(1, m + n + 1) options = options or {} return evalf(x, p, options)