# c#ython: profile=True
# remove extra # above for profiling

# Created by Peter Lepage (Cornell University) on 2011-08-17.
# Copyright (c) 2011-2012 G. Peter Lepage.
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# any later version (see <http://www.gnu.org/licenses/>).
# 
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.

import re
from ._svec_smat import svec, smat
from ._bufferdict import BufferDict

cimport numpy
cimport cython
from ._svec_smat cimport svec, smat


cdef extern from "math.h":
    double c_pow "pow" (double x,double y)
    double c_sin "sin" (double x)
    double c_cos "cos" (double x)
    double c_tan "tan" (double x)
    double c_sinh "sinh" (double x)
    double c_cosh "cosh" (double x)
    double c_tanh "tanh" (double x)
    double c_log "log" (double x)
    double c_exp "exp" (double x)
    double c_sqrt "sqrt" (double x)
    double c_asin "asin" (double x)
    double c_acos "acos" (double x)
    double c_atan "atan" (double x)


import numpy
from numpy import sin,cos,tan,exp,log,sqrt
from numpy import sinh,cosh,tanh,arcsin,arccos,arctan
from numpy import arcsinh,arccosh,arctanh
# from re import compile as _compile
import copy

try:
    import powerseries
    _ARRAY_TYPES = [numpy.ndarray,powerseries.PowerSeries]
except:
    _ARRAY_TYPES = [numpy.ndarray]
   

# GVar 
cdef class GVar:
    # cdef double v     -- value or mean
    # cdef svec d       -- vector of derivatives
    # cdef readonly smat cov    -- covariance matrix
    
    def __cinit__(self,double v,svec d,smat cov):
        self.v = v
        self.d = d
        self.cov = cov

    cpdef GVar clone(self):
        return GVar(self.v,self.d,self.cov)

    def __repr__(self):
        # return "construct_gvar(%s,%s,%s)" % (repr(self.mean),repr(self.der),repr(self.cov))
        return str(self)

    def __str__(self):
        """ Convert to string with format: mean +- std-dev. """
        return "%g +- %g"%(self.mean,self.sdev)

    def __hash__(self):
        return id(self)

    def __richcmp__(xx,yy,op):
        """ only == and != defined """
        if (op not in [2,3]):
            raise TypeError("unorderable types")
        if not (isinstance(xx,GVar) and isinstance(yy,GVar)):
            if not isinstance(xx, GVar):
                xx, yy = yy, xx
            if xx.sdev == 0 and xx.mean == yy:
                return True if op == 2 else False
            else:
                return False if op == 2 else True
        if ((xx.cov is yy.cov) and (xx.mean==yy.mean) 
                and numpy.all(xx.der==yy.der)):
            return True if op==2 else False
        else:
            return True if op==3 else False

    def __call__(self):
        """ Generate random number from ``self``'s distribution."""
        return numpy.random.normal(self.mean,self.sdev)

    def __neg__(self):
        return GVar(-self.v,self.d.mul(-1.),self.cov)

    def __pos__(self):
        return self

    def __add__(xx,yy):
        cdef GVar x,y
        cdef Py_ssize_t i,nx,di,ny
        if type(yy) in _ARRAY_TYPES:
            return NotImplemented   # let ndarray handle it
        elif isinstance(xx,GVar):
            if isinstance(yy,GVar):
                x = xx
                y = yy
                assert x.cov is y.cov,"Incompatible GVars."
                return GVar(x.v+y.v,x.d.add(y.d),x.cov)
            else:
                x = xx
                return GVar(x.v+yy,x.d,x.cov)
        elif isinstance(yy,GVar):
            y = yy
            return GVar(y.v+xx,y.d,y.cov)
        else: 
            return NotImplemented

    def __sub__(xx,yy):
        cdef GVar x,y
        if type(yy) in _ARRAY_TYPES:
            return NotImplemented   # let ndarray handle it
        elif isinstance(xx,GVar):
            if isinstance(yy,GVar):
                x = xx
                y = yy
                assert x.cov is y.cov,"Incompatible GVars."
                return GVar(x.v-y.v,x.d.add(y.d,1.,-1.),x.cov)
            else:
                x = xx
                return GVar(x.v-yy,x.d,x.cov)
        elif isinstance(yy,GVar):
            y = yy
            return GVar(xx-y.v,y.d.mul(-1.),y.cov)
        else: 
            return NotImplemented

    def __mul__(xx,yy):
        cdef GVar x,y
        
        if type(yy) in _ARRAY_TYPES:
            return NotImplemented   # let ndarray handle it
        elif isinstance(xx,GVar):
            if isinstance(yy,GVar):
                x = xx
                y = yy
                assert x.cov is y.cov,"Incompatible GVars."
                return GVar(x.v*y.v,x.d.add(y.d,y.v,x.v),x.cov)
            else:
                x = xx
                return GVar(x.v*yy,x.d.mul(yy),x.cov)
        elif isinstance(yy,GVar):
            y = yy
            return GVar(xx*y.v,y.d.mul(xx),y.cov)
        else: 
            return NotImplemented

    # truediv and div are the same --- 1st is for python3, 2nd for python2
    def __truediv__(xx,yy):
        cdef GVar x,y
        cdef double xd,yd
        if type(yy) in _ARRAY_TYPES:
            return NotImplemented   # let ndarray handle it
        elif isinstance(xx,GVar):
            if isinstance(yy,GVar):
                x = xx
                y = yy
                assert x.cov is y.cov,"Incompatible GVars."
                return GVar(x.v/y.v,x.d.add(y.d,1./y.v,-x.v/y.v**2),x.cov)
            else:
                x = xx
                yd=yy
                return GVar(x.v/yd,x.d.mul(1./yd),x.cov)
        elif isinstance(yy,GVar):
            y = yy
            xd=xx
            return GVar(xd/y.v,y.d.mul(-xd/y.v**2),y.cov)
        else: 
            return NotImplemented

    def __div__(xx,yy):
        cdef GVar x,y
        cdef double xd,yd
        if type(yy) in _ARRAY_TYPES:
            return NotImplemented   # let ndarray handle it
        elif isinstance(xx,GVar):
            if isinstance(yy,GVar):
                x = xx
                y = yy
                assert x.cov is y.cov,"Incompatible GVars."
                return GVar(x.v/y.v,x.d.add(y.d,1./y.v,-x.v/y.v**2),x.cov)
            else:
                x = xx
                yd=yy
                return GVar(x.v/yd,x.d.mul(1./yd),x.cov)
        elif isinstance(yy,GVar):
            y = yy
            xd=xx
            return GVar(xd/y.v,y.d.mul(-xd/y.v**2),y.cov)
        else: 
            return NotImplemented

    def __pow__(xx,yy,zz):
        cdef GVar x,y
        cdef double ans,f1,f2,yd,xd
        if type(yy) in _ARRAY_TYPES:
            return NotImplemented   # let ndarray handle it
        elif isinstance(xx,GVar):
            if isinstance(yy,GVar):
                x = xx
                y = yy
                assert x.cov is y.cov,"Incompatible GVars."
                ans = c_pow(x.v,y.v)
                f1 = c_pow(x.v,y.v-1)*y.v
                f2 = ans*c_log(x.v)
                return GVar(ans,x.d.add(y.d,f1,f2),x.cov)
            else:
                x = xx
                yd= yy
                ans = c_pow(x.v,yd)
                f1 = c_pow(x.v,yd-1)*yy
                return GVar(ans,x.d.mul(f1),x.cov)
        elif isinstance(yy,GVar):
            y = yy
            xd= xx
            ans = c_pow(xd,y.v)
            f1 = ans*c_log(xd)
            return GVar(ans,y.d.mul(f1),y.cov)
        else: 
            return NotImplemented

    def sin(self):
        return GVar(c_sin(self.v),self.d.mul(c_cos(self.v)),self.cov)

    def cos(self):
        return GVar(c_cos(self.v),self.d.mul(-c_sin(self.v)),self.cov)

    def tan(self):
        cdef double ans = c_tan(self.v)
        return GVar(ans,self.d.mul(1+ans*ans),self.cov)

    def arcsin(self):
        return GVar(c_asin(self.v),self.d.mul(1./(1.-self.v**2)**0.5),self.cov)

    def asin(self): 
        return self.arcsin()

    def arccos(self):
        return GVar(c_acos(self.v),self.d.mul(-1./(1.-self.v**2)**0.5),self.cov)

    def acos(self): 
        return self.arccos()

    def arctan(self):
        return GVar(c_atan(self.v),self.d.mul(1./(1.+self.v**2)),self.cov)

    def atan(self):
        return self.arctan()

    def sinh(self):
        return GVar(c_sinh(self.v),self.d.mul(c_cosh(self.v)),self.cov)

    def cosh(self):
        return GVar(c_cosh(self.v),self.d.mul(c_sinh(self.v)),self.cov)

    def tanh(self):
        return GVar(c_tanh(self.v),self.d.mul(1./(c_cosh(self.v)**2)),self.cov)

    def arcsinh(self):
        return log(self+sqrt(self*self+1.))

    def asinh(self):
        return self.arcsinh()

    def arccosh(self):
        return log(self+sqrt(self*self-1.))

    def acosh(self):
        return self.arccosh()

    def arctanh(self):
        return log((1.+self)/(1.-self))/2.

    def atanh(self):
        return self.arctanh()

    def exp(self):
        cdef double ans
        ans = c_exp(self.v)
        return GVar(ans,self.d.mul(ans),self.cov)

    def log(self):
        return GVar(c_log(self.v),self.d.mul(1./self.v),self.cov)
    
    def sqrt(self):
        cdef double ans = c_sqrt(self.v)
        return GVar(ans,self.d.mul(0.5/ans),self.cov)
    
    def deriv(GVar self, GVar x):
        """ Derivative of ``self`` with respest to *independent* |GVar| ``x``.

        ``x`` must be an *independent* |GVar|, which is a |GVar| created by a 
        call to :func:`gvar.gvar` (*e.g.*, ``x = gvar.gvar(xmean, xsdev)``) or a 
        function ``f(x)`` of such a |GVar|. (More precisely, ``x.der`` must have 
        only one nonzero entry.)

        All |GVar|\s are constructed from a set of independent |GVar|\s. 
        ``self.deriv(x)`` returns the partial derivative of ``self`` with 
        respect to independent |GVar| ``x``, holding all of the other 
        independent |GVar|\s constant.

        :param x: The independent |GVar|.
        :returns: The derivative of ``self`` with respect to ``x``.
        """
        cdef Py_ssize_t i, ider
        cdef double xder 
        xder = 0.0
        for i in range(x.d.size):
            if x.d.v[i].v != 0:
                if xder != 0:
                    raise ValueError("derivative ambiguous -- x is not independent")
                else:
                    xder = x.d.v[i].v
                    ider = x.d.v[i].i
        for i in range(self.d.size):
            if self.d.v[i].i == ider:
                return self.d.v[i].v / xder
        else:
            return 0.0

    def fmt(self, ndecimal=None, sep='', d=None):
        """ Convert to string with format: ``mean(sdev)``. 
            
        Leading zeros in the standard deviation are omitted: for example,
        ``25.67 +- 0.02`` becomes ``25.67(2)``. Parameter ``ndecimal``
        specifies how many digits follow the decimal point in the mean.
        Parameter ``sep`` is a string that is inserted between the ``mean``
        and the ``(sdev)``. If ``ndecimal`` is ``None`` (default), it is set
        automatically to the larger of ``int(2-log10(self.sdev))`` or
        ``0``; this will display at least two digits of error. Very large
        or very small numbers are written with exponential notation when
        ``ndecimal`` is ``None``.
        """
        if d is not None:
            ndecimal = d            # legacy name
        dv = self.sdev
        v = self.mean
        # special cases 
        if dv == float('inf'):
            # infinite sdev 
            if ndecimal is not None and ndecimal > 0:
                ft = "%%.%df" % int(ndecimal)
                return (ft % v) + ' +- inf'
            else:
                return str(v) + ' +- inf'

        if ndecimal is None:
            if dv == 0:
                #  simplify if self.sdev == 0 
                if v == 0:
                    return "0(0)"
                else:
                    ans = ("%g" % v).split('e')
                    if len(ans) == 2:
                        return ans[0] + "(0)e" + ans[1]
                    else:
                        return ans[0] + "(0)"

            if v == 0 or abs(dv/v) >= 100:
                #  set v=0 if self.sdev much larger than |self.mean| 
                if dv >= 1e6 or dv < 1e-5:
                    ans = ("%.1e" % dv).split('e')
                    return "0.0("+ans[0]+")e"+ans[1]
                else:
                    v = 0

            if abs(v/dv) >= 1e7:
                # use +- format if dv very small 
                return "%g +- %.2g" % (v,dv)

            if v != 0 and (abs(v) >= 1e6 or abs(v) < 1e-5):
                # exponential notation for large |self.mean| 
                exponent = numpy.floor(numpy.log10(abs(v)))
                fac = 10.**exponent
                mantissa = (self/fac).fmt(ndecimal)
                exponent = "e" + ("%.0e"%fac).split("e")[-1]
                return mantissa + exponent


        if ndecimal is None:
            # compute default number of decimal places 
            ndecimal = int(2-numpy.log10(dv))
            if ndecimal<0:
                ndecimal = 0
            elif round(dv*10.**ndecimal,0) == 100:
                ndecimal -= 1

        if ndecimal is None or ndecimal<0 or ndecimal!=int(ndecimal):
            # do not use compact notation 
            return self.__str__()

        v = round(v,ndecimal)
        dv = round(dv,ndecimal)
        if dv<1.0 and ndecimal>0:
            ft =  '%.'+str(ndecimal)+'f%s(%d)'
            dv = round(dv*10.**ndecimal,0)
            return ft % (v,sep,int(dv))
        else:
            ft = '%.'+str(ndecimal)+'f%s(%.'+str(ndecimal)+'f)'
            return ft % (v,sep,dv)

    def partialvar(self,*args):
        """ Compute partial variance due to |GVar|\s in ``args``.
            
        This method computes the part of ``self.var`` due to the |GVar|\s
        in ``args``. If ``args[i]`` is correlated with other |GVar|\s, the
        variance coming from these is included in the result as well. (This
        last convention is necessary because variances associated with
        correlated |GVar|\s cannot be disentangled into contributions
        corresponding to each variable separately.)
            
        :param args[i]: Variables contributing to the partial variance.
        :type args[i]: |GVar| or array/dictionary of |GVar|\s
        :returns: Partial variance due to all of ``args``.
        """
        cdef GVar ai
        cdef svec md
        cdef smat cov
        cdef numpy.ndarray[numpy.int_t,ndim=1] dmask
        cdef numpy.ndarray[numpy.int_t,ndim=1] md_idx
        cdef numpy.ndarray[numpy.double_t,ndim=1] md_v
        cdef Py_ssize_t i,j,md_size
        cdef Py_ssize_t dstart,dstop
        if self.d.size<=0:
            return 0.0
        dstart = self.d.v[0].i
        dstop = self.d.v[self.d.size-1].i+1
        # create a mask = 1 if (cov * args[i].der) component!=0; 0 otherwise 
        # a) collect all indices referenced in args[i].der 
        iset = set()
        for a in args:
            if a is None:
                continue
            if hasattr(a,'keys'):
                if not hasattr(a,'flat'):
                    a = BufferDict(a)
            else:
                a = numpy.asarray(a)
            for ai in a.flat:
                if ai is None:
                    continue
                else:
                    assert ai.cov is self.cov,"Incompatible |GVar|\s."
                iset.update(ai.d.indices())

        # b) collect indices connected to args[i].der indices by self.cov 
        cov = self.cov
        jset = set()
        for i in iset:
            jset.update(cov.rowlist[i].indices())

        # c) build the mask 
        dmask = numpy.zeros(dstop-dstart,int)
        for j in sorted(jset):
            if j<dstart:
                continue
            elif j>=dstop:
                break
            else:
                dmask[j-dstart] |= 1


        # create masked derivative vector for self 
        md_size = 0
        md_idx = numpy.zeros(dstop-dstart,int)
        md_v = numpy.zeros(dstop-dstart,float)
        for i in range(self.d.size):
            if dmask[self.d.v[i].i-dstart]==0:
                continue
            else:
                md_idx[md_size] = self.d.v[i].i
                md_v[md_size] = self.d.v[i].v
                md_size += 1
        md = svec(md_size)
        md._assign(md_v[:md_size],md_idx[:md_size])

        return md.dot(self.cov.dot(md))

    def partialsdev(self,*args): 
        """ Compute partial standard deviation due to |GVar|\s in ``args``.
            
        This method computes the part of ``self.sdev`` due to the |GVar|\s
        in ``args``. If ``args[i]`` is correlated with other |GVar|\s, the
        standard deviation coming from these is included in the result as
        well. (This last convention is necessary because variances
        associated with correlated |GVar|\s cannot be disentangled into
        contributions corresponding to each variable separately.)
            
        :param args[i]: Variables contributing to the partial standard
            deviation.
        :type args[i]: |GVar| or array/dictionary of |GVar|\s
        :returns: Partial standard deviation due to ``args``.
        """
        ans = self.partialvar(*args)
        return ans**0.5 if ans>0 else -(-ans)**0.5

    property val:
        """ Mean value. """ 
        def __get__(self):
            return self.v

    property der:
        """ Array of derivatives with respect to  underlying (original)
        |GVar|\s. 
        """
        def __get__(self):
            return self.d.toarray(len(self.cov))

    property mean:
        """ Mean value. """
        def __get__(self):
            return self.v

    property sdev:
        """ Standard deviation. """
        def __get__(self):
            cdef double var = self.cov.expval(self.d) 
            if var>=0:
                return c_sqrt(var)
            else:
                return -c_sqrt(-var)

    property var:
        """ Variance. """
        # @cython.boundscheck(False)
        def __get__(self):
            return self.cov.expval(self.d)  

    def dotder(self,numpy.ndarray[numpy.double_t,ndim=1] v not None):
        """ Return the dot product of ``self.der`` and ``v``. """
        cdef double ans = 0
        cdef Py_ssize_t i
        for i in range(self.d.size):
            ans += v[self.d.v[i].i]*self.d.v[i].v
        return ans




# GVar factory functions 
    
_RE1 = re.compile(r"(.*)\s*[+][-]\s*(.*)")
_RE2 = re.compile(r"(.*)[e](.*)")
_RE3 = re.compile(r"([-+]?)([0-9]*)[.]?([0-9]*)\s*\(([0-9]+)\)")
_RE3a = re.compile(r"([-+]?[0-9]*[.]?[0-9]*)\s*\(([.0-9]+)\)")
                       
class GVarFactory:
    """ Create one or more new |GVar|\s.
        
    Each of the following creates new |GVar|\s:
        
    .. function:: gvar(x,xsdev)
        
        Returns a |GVar| with mean ``x`` and standard deviation ``xsdev``.
        Returns an array of |GVar|\s if ``x`` and ``xsdev`` are arrays
        with the same shape; the shape of the result is the same as the
        shape of ``x``. 
        
    .. function:: gvar(x,xcov)
        
        Returns an array of |GVar|\s with means given by array ``x`` and a
        covariance matrix given by array ``xcov``, where ``xcov.shape =
        2*x.shape``. The result has the same shape as ``x``.
        
    .. function:: gvar((x,xsdev))
        
        Returns a |GVar| with mean ``x`` and standard deviation ``xsdev``.
        
    .. function:: gvar(xstr)
        
        Returns a |GVar| corresponding to string ``xstr`` which is 
        either of the form ``"xmean +- xsdev"`` or ``"x(xerr)"`` (see
        :meth:`GVar.fmt`).
        
    .. function:: gvar(xgvar)
        
        Returns |GVar| ``xgvar`` unchanged.
        
    .. function:: gvar(xdict)
        
        Returns a dictionary (:class:`BufferDict`) ``b`` where 
        ``b[k] = gvar(xdict[k])`` for every key in dictionary ``xdict``.
        The values in ``xdict``, therefore, can be strings, tuples or 
        |GVar|\s (see above), or arrays of these.
        
    .. function:: gvar(xarray)
        
        Returns an array ``a`` having the same shape as ``xarray`` where
        every element ``a[i...] = gvar(xarray[i...])``. The values in
        ``xarray``, therefore, can be strings, tuples or |GVar|\s (see
        above).
                    
    ``gvar.gvar`` is actually an object of type :class:`gvar.GVarFactory`.  
    """
    def __init__(self,cov=None):
        if cov is None:
            self.cov = smat()
        else:
            assert isinstance(cov,smat),"cov not type gvar.smat"
            self.cov = cov

    def __call__(self,*args):
        cdef Py_ssize_t nx,i,nd
        cdef svec der
        cdef smat cov
        cdef GVar gd
        cdef numpy.ndarray[numpy.double_t,ndim=1] d
        cdef numpy.ndarray[numpy.double_t,ndim=1] d_v
        cdef numpy.ndarray[numpy.int_t,ndim=1] d_idx
        
        if len(args)==2:
            # (x,xsdev) or (xarray,sdev-array) or (xarray,cov) 
            # unpack arguments and verify types 
            try:
                x = numpy.asarray(args[0],float)
                xsdev = numpy.asarray(args[1],float)
            except (ValueError,TypeError):
                raise TypeError(    #):
                        "Arguments must be numbers or arrays of numbers")

            if len(x.shape)==0:
                # single gvar from x and xsdev 
                if len(xsdev.shape)!=0:
                    raise ValueError("x and xsdev different shapes.")
                idx = self.cov.append_diag(numpy.array([xsdev**2]))
                der = svec(1)
                der.v[0].i = idx[0]
                der.v[0].v = 1.0
                return GVar(x,der,self.cov)

            else:
                # array of gvars from x and sdev/cov arrays 
                nx = len(x.flat)
                if x.shape==xsdev.shape:  # x,sdev
                    idx = self.cov.append_diag(xsdev.reshape(nx)**2)
                elif xsdev.shape==2*x.shape: # x,cov
                    idx = self.cov.append_diag_m(xsdev.reshape(nx,nx))
                else:
                    raise ValueError("Argument shapes mismatched: "+
                        str(x.shape)+' '+str(xsdev.shape))
                d = numpy.ones(nx,float)
                ans = []
                for i in range(nx):
                    der = svec(1)
                    der.v[0].i = idx[i]
                    der.v[0].v = 1.0
                    ans.append(GVar(x.flat[i],der,self.cov))
                return numpy.array(ans).reshape(x.shape)


        elif len(args)==1:
            x = args[0]
            if isinstance(x,str):
                # case 1: x is a string like "3.72(41)" or "3.2 +- 4" 
                x = x.strip()
                try:
                    # eg: 3.4 +- 0.7e-4
                    a,c = _RE1.match(x).groups()
                    return self(float(a), float(c))
                except AttributeError:
                    pass
                try:
                    # eg: 3.4(1)e+10
                    a,c = _RE2.match(x).groups()
                    return self(a)*float("1e"+c)
                except AttributeError:
                    pass
                try:
                    # eg: +3.456(33)
                    s,a,b,c = _RE3.match(x).groups()
                    s = -1. if s == '-' else 1.
                    if not a and not b:
                        raise ValueError("Poorly formatted string: "+x)
                    elif not b:
                        return s*self(float(a),float(c))
                    else:
                        if not a:
                            a = '0'
                        fac = 1./10.**len(b)
                        a,b,c = [float(xi) for xi in [a,b,c]]
                        return s*self(a + b*fac, c*fac)
                except AttributeError:
                    pass
                try:
                    # eg: 3.456(1.234)
                    a,c = _RE3a.match(x).groups()
                    return self(float(a), float(c))
                except AttributeError:
                    raise ValueError("Poorly formatted string: "+x)

            elif isinstance(x,GVar):
                # case 2: x is a GVar 
                return x

            elif isinstance(x,tuple) and len(x)==2:
                # case 3: x = (x,sdev) tuple 
                return self(x[0],x[1])

            elif hasattr(x,'keys'):
                # case 4: x is a dictionary 
                ans = BufferDict()
                for k in x:
                    ans[k] = self(x[k])
                return ans

            elif hasattr(x,'__iter__'):
                # case 5: x is an array 
                try:
                    xa = numpy.asarray(x)
                except ValueError:
                    xa = numpy.asarray(x,object)
                if xa.size==0:
                    return xa
                if xa.shape != () and xa.shape[-1]==2 and xa.dtype!=object and xa.ndim>1:
                    # array of tuples? 
                    xxa = numpy.empty(xa.shape[:-1],object)
                    xxa[:] = x
                    if all(type(xxai)==tuple for xxai in xxa.flat):
                        return self(xa[...,0],xa[...,1])

                return numpy.array([xai if isinstance(xai,GVar) else self(xai) 
                                    for xai in xa.flat]).reshape(xa.shape)

            else:   # case 6: a number
                return self(x,0.0)
        elif len(args)==3:
            # (x,der,cov) 
            try:
                x = numpy.asarray(args[0],float)
                d = numpy.asarray(args[1],float).flatten()
            except (ValueError,TypeError,AssertionError):
                raise TypeError("Value and derivatives not numbers.")
            assert len(x.shape)==0,"Value not a number."
            cov = args[2]
            assert isinstance(cov,smat),"cov not type gvar.smat."
            assert len(cov)>=len(d),"length mismatch between d and cov"
            d_idx = d.nonzero()[0]
            d_v = d[d_idx]
            nd = len(d_idx)
            der = svec(nd)
            for i in range(nd):
                der.v[i].i = d_idx[i]
                der.v[i].v = d_v[i]
            return GVar(x,der,cov)

        else:
            raise ValueError("Wrong number of arguments: "+str(len(args)))


        


 