Conversion of symbolic expressions to other types

This module provides routines for converting new symbolic expressions to other types. Primarily, it provides a class Converter which will walk the expression tree and make calls to methods overridden by subclasses.

class sage.symbolic.expression_conversions.AlgebraicConverter(field)

Bases: sage.symbolic.expression_conversions.Converter

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: a = AlgebraicConverter(QQbar)
sage: a.field
Algebraic Field
sage: a.reciprocal_trig_functions['cot']
tan
arithmetic(ex, operator)

Convert a symbolic expression to an algebraic number.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: f = 2^(1/2)
sage: a = AlgebraicConverter(QQbar)
sage: a.arithmetic(f, f.operator())
1.414213562373095?

TESTS:

sage: f = pi^6
sage: a = AlgebraicConverter(QQbar)
sage: a.arithmetic(f, f.operator())
Traceback (most recent call last):
...
TypeError: unable to convert pi^6 to Algebraic Field
composition(ex, operator)

Coerce to an algebraic number.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: a = AlgebraicConverter(QQbar)
sage: a.composition(exp(I*pi/3), exp)
0.500000000000000? + 0.866025403784439?*I
sage: a.composition(sin(pi/5), sin)
0.5877852522924731? + 0.?e-18*I

TESTS:

sage: QQbar(zeta(7))
Traceback (most recent call last):
...
TypeError: unable to convert zeta(7) to Algebraic Field
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: a = AlgebraicConverter(QQbar)
sage: f = SR(2)
sage: a.pyobject(f, f.pyobject())
2
sage: _.parent()
Algebraic Field
class sage.symbolic.expression_conversions.Converter(use_fake_div=False)

Bases: object

If use_fake_div is set to True, then the converter will try to replace expressions whose operator is operator.mul with the corresponding expression whose operator is operator.div.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import Converter
sage: c = Converter(use_fake_div=True)
sage: c.use_fake_div
True
arithmetic(ex, operator)

The input to this method is a symbolic expression and the infix operator corresponding to that expression. Typically, one will convert all of the arguments and then perform the operation afterward.

TESTS:

sage: from sage.symbolic.expression_conversions import Converter
sage: f = x + 2
sage: Converter().arithmetic(f, f.operator())
Traceback (most recent call last):
...
NotImplementedError: arithmetic
composition(ex, operator)

The input to this method is a symbolic expression and its operator. This method will get called when you have a symbolic function application.

TESTS:

sage: from sage.symbolic.expression_conversions import Converter
sage: f = sin(2)
sage: Converter().composition(f, f.operator())
Traceback (most recent call last):
...
NotImplementedError: composition
derivative(ex, operator)

The input to this method is a symbolic expression which corresponds to a relation.

TESTS:

sage: from sage.symbolic.expression_conversions import Converter
sage: a = function('f', x).diff(x); a
D[0](f)(x)
sage: Converter().derivative(a, a.operator())
Traceback (most recent call last):
...
NotImplementedError: derivative
get_fake_div(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import Converter
sage: c = Converter(use_fake_div=True)
sage: c.get_fake_div(sin(x)/x)
FakeExpression([sin(x), x], <built-in function div>)
sage: c.get_fake_div(-1*sin(x))
FakeExpression([sin(x)], <built-in function neg>)
sage: c.get_fake_div(-x)
FakeExpression([x], <built-in function neg>)
sage: c.get_fake_div((2*x^3+2*x-1)/((x-2)*(x+1)))
FakeExpression([2*x^3 + 2*x - 1, FakeExpression([x + 1, x - 2], <built-in function mul>)], <built-in function div>)

Check if #8056 is fixed, i.e., if numerator is 1.:

sage: c.get_fake_div(1/pi/x)
FakeExpression([1, FakeExpression([pi, x], <built-in function mul>)], <built-in function div>)
pyobject(ex, obj)

The input to this method is the result of calling pyobject() on a symbolic expression.

Note

Note that if a constant such as pi is encountered in the expression tree, its corresponding pyobject which is an instance of sage.symbolic.constants.Pi will be passed into this method. One cannot do arithmetic using such an object.

TESTS:

sage: from sage.symbolic.expression_conversions import Converter
sage: f = SR(1)
sage: Converter().pyobject(f, f.pyobject())
Traceback (most recent call last):
...
NotImplementedError: pyobject
relation(ex, operator)

The input to this method is a symbolic expression which corresponds to a relation.

TESTS:

sage: from sage.symbolic.expression_conversions import Converter
sage: import operator
sage: Converter().relation(x==3, operator.eq)
Traceback (most recent call last):
...
NotImplementedError: relation
sage: Converter().relation(x==3, operator.lt)
Traceback (most recent call last):
...
NotImplementedError: relation
symbol(ex)

The input to this method is a symbolic expression which corresponds to a single variable. For example, this method could be used to return a generator for a polynomial ring.

TESTS:

sage: from sage.symbolic.expression_conversions import Converter
sage: Converter().symbol(x)
Traceback (most recent call last):
...
NotImplementedError: symbol
class sage.symbolic.expression_conversions.FakeExpression(operands, operator)

Bases: object

Pynac represents \(x/y\) as \(xy^{-1}\). Often, tree-walkers would prefer to see divisions instead of multiplications and negative exponents. To allow for this (since Pynac internally doesn’t have division at all), there is a possibility to pass use_fake_div=True; this will rewrite an Expression into a mixture of Expression and FakeExpression nodes, where the FakeExpression nodes are used to represent divisions. These nodes are intended to act sufficiently like Expression nodes that tree-walkers won’t care about the difference.

operands()

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FakeExpression
sage: import operator; x,y = var('x,y')
sage: f = FakeExpression([x, y], operator.div)
sage: f.operands()
[x, y]
operator()

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FakeExpression
sage: import operator; x,y = var('x,y')
sage: f = FakeExpression([x, y], operator.div)
sage: f.operator()
<built-in function div>
pyobject()

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FakeExpression
sage: import operator; x,y = var('x,y')
sage: f = FakeExpression([x, y], operator.div)
sage: f.pyobject()
Traceback (most recent call last):
...
TypeError: self must be a numeric expression
class sage.symbolic.expression_conversions.FastCallableConverter(ex, etb)

Bases: sage.symbolic.expression_conversions.Converter

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FastCallableConverter
sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x'])
sage: f = FastCallableConverter(x+2, etb)
sage: f.ex
x + 2
sage: f.etb
<sage.ext.fast_callable.ExpressionTreeBuilder object at 0x...>
sage: f.use_fake_div
True
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: var('x,y')
(x, y)
sage: (x+y)._fast_callable_(etb)
add(v_0, v_1)
sage: (-x)._fast_callable_(etb)
neg(v_0)
sage: (x+y+x^2)._fast_callable_(etb)
add(add(ipow(v_0, 2), v_0), v_1)

TESTS:

Check if rational functions with numerator 1 can be converted. #8056:

sage: (1/pi/x)._fast_callable_(etb)
div(1, mul(pi, v_0))

sage: etb = ExpressionTreeBuilder(vars=['x'], domain=RDF)
sage: (x^7)._fast_callable_(etb)
ipow(v_0, 7)
sage: f(x)=1/pi/x; plot(f,2,3)
composition(ex, function)

Given an ExpressionTreeBuilder, return an Expression representing this value.

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: x,y = var('x,y')
sage: sin(sqrt(x+y))._fast_callable_(etb)
sin(sqrt(add(v_0, v_1)))
sage: arctan2(x,y)._fast_callable_(etb)
{arctan2}(v_0, v_1)
pyobject(ex, obj)

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x'])
sage: pi._fast_callable_(etb)
pi
sage: etb = ExpressionTreeBuilder(vars=['x'], domain=RDF)
sage: pi._fast_callable_(etb)
3.14159265359
relation(ex, operator)

EXAMPLES:

sage: ff = fast_callable(x == 2, vars=['x'])
sage: ff(2)
0
sage: ff(4)
2
sage: ff = fast_callable(x < 2, vars=['x'])
Traceback (most recent call last):
...
NotImplementedError
symbol(ex)

Given an ExpressionTreeBuilder, return an Expression representing this value.

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: x, y, z = var('x,y,z')
sage: x._fast_callable_(etb)
v_0
sage: y._fast_callable_(etb)
v_1
sage: z._fast_callable_(etb)
Traceback (most recent call last):
...
ValueError: Variable 'z' not found
class sage.symbolic.expression_conversions.FastFloatConverter(ex, *vars)

Bases: sage.symbolic.expression_conversions.Converter

Returns an object which provides fast floating point evaluation of the symbolic expression ex. This is an class used internally and is not meant to be used directly.

See sage.ext.fast_eval for more information.

EXAMPLES:

sage: x,y,z = var('x,y,z')
sage: f = 1 + sin(x)/x + sqrt(z^2+y^2)/cosh(x)
sage: ff = f._fast_float_('x', 'y', 'z')
sage: f(x=1.0,y=2.0,z=3.0).n()
4.1780638977...
sage: ff(1.0,2.0,3.0)
4.1780638977...

Using _fast_float_ without specifying the variable names is deprecated:

sage: f = x._fast_float_()
doctest:...: DeprecationWarning: Substitution using
function-call syntax and unnamed arguments is deprecated
and will be removed from a future release of Sage; you
can use named arguments instead, like EXPR(x=..., y=...)
See http://trac.sagemath.org/5930 for details.
sage: f(1.2)
1.2

Using _fast_float_ on a function which is the identity is now supported (see Trac 10246):

sage: f = symbolic_expression(x).function(x)
sage: f._fast_float_(x)
<sage.ext.fast_eval.FastDoubleFunc object at ...>
sage: f(22)
22
arithmetic(ex, operator)

EXAMPLES:

sage: x,y = var('x,y')
sage: f = x*x-y
sage: ff = f._fast_float_('x','y')
sage: ff(2,3)
1.0

sage: a = x + 2*y
sage: f = a._fast_float_('x', 'y')
sage: f(1,0)
1.0
sage: f(0,1)
2.0

sage: f = sqrt(x)._fast_float_('x'); f.op_list()
['load 0', 'call sqrt(1)']

sage: f = (1/2*x)._fast_float_('x'); f.op_list()
['load 0', 'push 0.5', 'mul']
composition(ex, operator)

EXAMPLES:

sage: f = sqrt(x)._fast_float_('x')
sage: f(2)
1.41421356237309...
sage: y = var('y')
sage: f = sqrt(x+y)._fast_float_('x', 'y')
sage: f(1,1)
1.41421356237309...
sage: f = sqrt(x+2*y)._fast_float_('x', 'y')
sage: f(2,0)
1.41421356237309...
sage: f(0,1)
1.41421356237309...
pyobject(ex, obj)

EXAMPLES:

sage: f = SR(2)._fast_float_()
sage: f(3)
2.0
relation(ex, operator)

EXAMPLES:

sage: ff = fast_float(x == 2, 'x')
sage: ff(2)
0.0
sage: ff(4)
2.0
sage: ff = fast_float(x < 2, 'x')
Traceback (most recent call last):
...
NotImplementedError
symbol(ex)

EXAMPLES:

sage: f = x._fast_float_('x', 'y')
sage: f(1,2)
1.0
sage: f = x._fast_float_('y', 'x')
sage: f(1,2)
2.0
class sage.symbolic.expression_conversions.InterfaceInit(interface)

Bases: sage.symbolic.expression_conversions.Converter

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: a = pi + 2
sage: m(a)
'(%pi)+(2)'
sage: m(sin(a))
'sin((%pi)+(2))'
sage: m(exp(x^2) + pi + 2)
'(%pi)+(exp((x)^(2)))+(2)'
arithmetic(ex, operator)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.arithmetic(x+2, operator.add)
'(x)+(2)'
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.composition(sin(x), sin)
'sin(x)'
sage: m.composition(ceil(x), ceil)
'ceiling(x)'

sage: m = InterfaceInit(mathematica)
sage: m.composition(sin(x), sin)
'Sin[x]'
derivative(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: f = function('f')
sage: a = f(x).diff(x); a
D[0](f)(x)
sage: print m.derivative(a, a.operator())
diff('f(x), x, 1)
sage: b = f(x).diff(x, x)
sage: print m.derivative(b, b.operator())
diff('f(x), x, 2)

We can also convert expressions where the argument is not just a variable, but the result is an “at” expression using temporary variables:

sage: y = var('y')
sage: t = (f(x*y).diff(x))/y
sage: t
D[0](f)(x*y)
sage: m.derivative(t, t.operator())
"at(diff('f(t0), t0, 1), [t0 = x*y])"
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: ii = InterfaceInit(gp)
sage: f = 2+I
sage: ii.pyobject(f, f.pyobject())
'I + 2'

sage: ii.pyobject(SR(2), 2)
'2'

sage: ii.pyobject(pi, pi.pyobject())
'Pi'
relation(ex, operator)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.relation(x==3, operator.eq)
'x = 3'
sage: m.relation(x==3, operator.lt)
'x < 3'
symbol(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.symbol(x)
'x'
sage: f(x) = x
sage: m.symbol(f)
'x'
class sage.symbolic.expression_conversions.PolynomialConverter(ex, base_ring=None, ring=None)

Bases: sage.symbolic.expression_conversions.Converter

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: x, y = var('x,y')
sage: p = PolynomialConverter(x+y, base_ring=QQ)
sage: p.base_ring
Rational Field
sage: p.ring
Multivariate Polynomial Ring in x, y over Rational Field

sage: p = PolynomialConverter(x, base_ring=QQ)
sage: p.base_ring
Rational Field
sage: p.ring
Univariate Polynomial Ring in x over Rational Field

sage: p = PolynomialConverter(x, ring=QQ['x,y'])
sage: p.base_ring
Rational Field
sage: p.ring
Multivariate Polynomial Ring in x, y over Rational Field

sage: p = PolynomialConverter(x+y, ring=QQ['x'])
Traceback (most recent call last):
...
TypeError: y is not a variable of Univariate Polynomial Ring in x over Rational Field
arithmetic(ex, operator)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import PolynomialConverter

sage: x, y = var('x, y')
sage: p = PolynomialConverter(x, base_ring=RR)
sage: p.arithmetic(pi+e, operator.add)
5.85987448204884
sage: p.arithmetic(x^2, operator.pow)
x^2

sage: p = PolynomialConverter(x+y, base_ring=RR)
sage: p.arithmetic(x*y+y^2, operator.add)
x*y + y^2

sage: p = PolynomialConverter(y^(3/2), ring=SR['x'])
sage: p.arithmetic(y^(3/2), operator.pow)
y^(3/2)
sage: _.parent()
Symbolic Ring
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: a = sin(2)
sage: p = PolynomialConverter(a*x, base_ring=RR)
sage: p.composition(a, a.operator())
0.909297426825682
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: p = PolynomialConverter(x, base_ring=QQ)
sage: f = SR(2)
sage: p.pyobject(f, f.pyobject())
2
sage: _.parent()
Rational Field
relation(ex, op)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import PolynomialConverter

sage: x, y = var('x, y')
sage: p = PolynomialConverter(x, base_ring=RR)

sage: p.relation(x==3, operator.eq)
x - 3.00000000000000
sage: p.relation(x==3, operator.lt)
Traceback (most recent call last):
...
ValueError: Unable to represent as a polynomial

sage: p = PolynomialConverter(x - y, base_ring=QQ)
sage: p.relation(x^2 - y^3 + 1 == x^3, operator.eq)
-x^3 - y^3 + x^2 + 1
symbol(ex)

Returns a variable in the polynomial ring.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: p = PolynomialConverter(x, base_ring=QQ)
sage: p.symbol(x)
x
sage: _.parent()
Univariate Polynomial Ring in x over Rational Field
class sage.symbolic.expression_conversions.RingConverter(R, subs_dict=None)

Bases: sage.symbolic.expression_conversions.Converter

A class to convert expressions to other rings.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF, subs_dict={x:2})
sage: R.ring
Real Interval Field with 53 bits of precision
sage: R.subs_dict
{x: 2}
sage: R(pi+e)
5.85987448204884?
sage: loads(dumps(R))
<sage.symbolic.expression_conversions.RingConverter object at 0x...>
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: P.<z> = ZZ[]
sage: R = RingConverter(P, subs_dict={x:z})
sage: a = 2*x^2 + x + 3
sage: R(a)
2*z^2 + z + 3
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF)
sage: R(cos(2))
-0.4161468365471424?
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF)
sage: R(SR(5/2))
2.5000000000000000?
symbol(ex)

All symbols appearing in the expression must appear in subs_dict in order for the conversion to be successful.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF, subs_dict={x:2})
sage: R(x+pi)
5.141592653589794?

sage: R = RingConverter(RIF)
sage: R(x+pi)
Traceback (most recent call last):
...
TypeError
class sage.symbolic.expression_conversions.SubstituteFunction(ex, original, new)

Bases: sage.symbolic.expression_conversions.Converter

A class that walks the tree and replaces occurrences of a function with another.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: s(1/foo(foo(x)) + foo(2))
1/bar(bar(x)) + bar(2)
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: f = x*foo(x) + pi/foo(x)
sage: s.arithmetic(f, f.operator())
x*bar(x) + pi/bar(x)
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: f = foo(x)
sage: s.composition(f, f.operator())
bar(x)
sage: f = foo(foo(x))
sage: s.composition(f, f.operator())
bar(bar(x))
sage: f = sin(foo(x))
sage: s.composition(f, f.operator())
sin(bar(x))
sage: f = foo(sin(x))
sage: s.composition(f, f.operator())
bar(sin(x))
derivative(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: f = foo(x).diff(x)
sage: s.derivative(f, f.operator())
D[0](bar)(x)

TESTS:

We can substitute functions under a derivative operator, trac ticket #12801:

sage: f = function('f')
sage: g = function('g')
sage: f(g(x)).diff(x).substitute_function(g, sin)
cos(x)*D[0](f)(sin(x))
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: f = SR(2)
sage: s.pyobject(f, f.pyobject())
2
sage: _.parent()
Symbolic Ring
relation(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: eq = foo(x) == x
sage: s.relation(eq, eq.operator())
bar(x) == x
symbol(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: s.symbol(x)
x
class sage.symbolic.expression_conversions.SympyConverter(use_fake_div=False)

Bases: sage.symbolic.expression_conversions.Converter

Converts any expression to SymPy.

EXAMPLE:

sage: import sympy
sage: var('x,y')
(x, y)
sage: f = exp(x^2) - arcsin(pi+x)/y
sage: f._sympy_()
exp(x**2) - asin(x + pi)/y
sage: _._sage_()
-arcsin(pi + x)/y + e^(x^2)

sage: sympy.sympify(x) # indirect doctest
x

TESTS:

Make sure we can convert I (trac #6424):

sage: bool(I._sympy_() == I)
True
sage: (x+I)._sympy_()
x + I
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: f = x + 2
sage: s.arithmetic(f, f.operator())
x + 2
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: f = sin(2)
sage: s.composition(f, f.operator())
sin(2)
sage: type(_)
sin
sage: f = arcsin(2)
sage: s.composition(f, f.operator())
asin(2)
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: f = SR(2)
sage: s.pyobject(f, f.pyobject())
2
sage: type(_)
<class 'sympy.core.numbers.Integer'>
symbol(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: s.symbol(x)
x
sage: type(_)
<class 'sympy.core.symbol.Symbol'>
sage.symbolic.expression_conversions.algebraic(ex, field)

Returns the symbolic expression ex as a element of the algebraic field field.

EXAMPLES:

sage: a = SR(5/6)
sage: AA(a)
5/6
sage: type(AA(a))
<class 'sage.rings.qqbar.AlgebraicReal'>
sage: QQbar(a)
5/6
sage: type(QQbar(a))
<class 'sage.rings.qqbar.AlgebraicNumber'>
sage: QQbar(i)
1*I
sage: AA(golden_ratio)
1.618033988749895?
sage: QQbar(golden_ratio)
1.618033988749895?
sage: QQbar(sin(pi/3))
0.866025403784439?

sage: QQbar(sqrt(2) + sqrt(8))
4.242640687119285?
sage: AA(sqrt(2) ^ 4) == 4
True
sage: AA(-golden_ratio)
-1.618033988749895?
sage: QQbar((2*I)^(1/2))
1 + 1*I
sage: QQbar(e^(pi*I/3))
0.500000000000000? + 0.866025403784439?*I

sage: AA(x*sin(0))
0
sage: QQbar(x*sin(0))
0
sage.symbolic.expression_conversions.fast_callable(ex, etb)

Given an ExpressionTreeBuilder etb, return an Expression representing the symbolic expression ex.

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: x,y = var('x,y')
sage: f = y+2*x^2
sage: f._fast_callable_(etb)
add(mul(ipow(v_0, 2), 2), v_1)

sage: f = (2*x^3+2*x-1)/((x-2)*(x+1))
sage: f._fast_callable_(etb)
div(add(add(mul(ipow(v_0, 3), 2), mul(v_0, 2)), -1), mul(add(v_0, 1), add(v_0, -2)))
sage.symbolic.expression_conversions.fast_float(ex, *vars)

Returns an object which provides fast floating point evaluation of the symbolic expression ex.

See sage.ext.fast_eval for more information.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import fast_float
sage: f = sqrt(x+1)
sage: ff = fast_float(f, 'x')
sage: ff(1.0)
1.4142135623730951
sage.symbolic.expression_conversions.polynomial(ex, base_ring=None, ring=None)

Returns a polynomial from the symbolic expression ex. Either a base ring base_ring or a polynomial ring ring can be specified for the parent of result. If just a base ring is given, then the variables of the base ring will be the variables of the expression ex.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import polynomial
sage: f = x^2 + 2
sage: polynomial(f, base_ring=QQ)
x^2 + 2
sage: _.parent()
Univariate Polynomial Ring in x over Rational Field

sage: polynomial(f, ring=QQ['x,y'])
x^2 + 2
sage: _.parent()
Multivariate Polynomial Ring in x, y over Rational Field

sage: x, y = var('x, y')
sage: polynomial(x + y^2, ring=QQ['x,y'])
y^2 + x
sage: _.parent()
Multivariate Polynomial Ring in x, y over Rational Field

sage: s,t=var('s,t')
sage: expr=t^2-2*s*t+1
sage: expr.polynomial(None,ring=SR['t'])
t^2 - 2*s*t + 1
sage: _.parent()
Univariate Polynomial Ring in t over Symbolic Ring

sage: polynomial(y - sqrt(x), ring=SR[y])
y - sqrt(x)
sage: _.list()
[-sqrt(x), 1]

The polynomials can have arbitrary (constant) coefficients so long as they coerce into the base ring:

sage: polynomial(2^sin(2)*x^2 + exp(3), base_ring=RR)
1.87813065119873*x^2 + 20.0855369231877

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