The Sage Preparser#



sage: preparse('2/3')
sage: preparse('2.5')
sage: preparse('2^3')
sage: preparse('a^b')            # exponent
sage: preparse('a**b')
sage: preparse('G.0')            # generator
sage: preparse('a = 939393R')    # raw
'a = 939393'
sage: implicit_multiplication(True)
sage: preparse('a b c in L')     # implicit multiplication
'a*b*c in L'
sage: preparse('2e3x + 3exp(y)')
"RealNumber('2e3')*x + Integer(3)*exp(y)"

A string with escaped quotes in it (the point here is that the preparser does not get confused by the internal quotes):

sage: ""Yes," he said."
'"Yes," he said.'
sage: s = "\"; s

A hex literal:

sage: preparse('0x2e3')
sage: 0xA
sage: 0xe

Raw and hex work correctly:

sage: type(0xa1)
<class 'sage.rings.integer.Integer'>
sage: type(0xa1r)
<class 'int'>
sage: type(0Xa1R)
<class 'int'>

The preparser can handle PEP 515 (see github issue #28490):

sage: 1_000_000 + 3_000

In Sage, methods can also be called on integer and real literals (note that in pure Python this would be a syntax error):

sage: 16.sqrt()
sage: 87.factor()
3 * 29
sage: 15.10.sqrt()
sage: preparse('87.sqrt()')
sage: preparse('15.10.sqrt()')

Note that calling methods on int literals in pure Python is a syntax error, but Sage allows this for Sage integers and reals, because users frequently request it:

sage: eval('4.__add__(3)')
Traceback (most recent call last):
SyntaxError: invalid ...

Symbolic functional notation:

sage: # needs sage.symbolic
sage: a=10; f(theta, beta) = theta + beta; b = x^2 + theta
sage: f
(theta, beta) |--> beta + theta
sage: a
sage: b
x^2 + theta
sage: f(theta,theta)

sage: a = 5; f(x,y) = x*y*sqrt(a)                                                   # needs sage.symbolic
sage: f                                                                             # needs sage.symbolic
(x, y) |--> sqrt(5)*x*y

This involves an =-, but should still be turned into a symbolic expression:

sage: preparse('a(x) =- 5')
'__tmp__=var("x"); a = symbolic_expression(- Integer(5)).function(x)'
sage: f(x)=-x                                                                       # needs sage.symbolic
sage: f(10)                                                                         # needs sage.symbolic

This involves -=, which should not be turned into a symbolic expression (of course a(x) is not an identifier, so this will never be valid):

sage: preparse('a(x) -= 5')
'a(x) -= Integer(5)'

Raw literals:

Raw literals are not preparsed, which can be useful from an efficiency point of view. Just like Python ints are denoted by an L, in Sage raw integer and floating literals are followed by an”r” (or “R”) for raw, meaning not preparsed.

We create a raw integer:

sage: a = 393939r
sage: a
sage: type(a)
<class 'int'>

We create a raw float:

sage: z = 1.5949r
sage: z
sage: type(z)
<class 'float'>

You can also use an upper case letter:

sage: z = 3.1415R
sage: z
sage: type(z)
<class 'float'>

This next example illustrates how raw literals can be very useful in certain cases. We make a list of even integers up to 10000:

sage: v = [ 2*i for i in range(10000)]

This takes a noticeable fraction of a second (e.g., 0.25 seconds). After preparsing, what Python is really executing is the following:

sage: preparse('v = [ 2*i for i in range(10000)]')
'v = [ Integer(2)*i for i in range(Integer(10000))]'

If instead we use a raw 2 we get execution that is instant (0.00 seconds):

sage: v = [ 2r * i for i in range(10000r)]

Behind the scenes what happens is the following:

sage: preparse('v = [ 2r * i for i in range(10000r)]')
'v = [ 2 * i for i in range(10000)]'


The results of the above two expressions are different. The first one computes a list of Sage integers, whereas the second creates a list of Python integers. Python integers are typically much more efficient than Sage integers when they are very small; large Sage integers are much more efficient than Python integers, since they are implemented using the GMP C library.

F-Strings (PEP 498):

Expressions embedded within F-strings are preparsed:

sage: f'{1/3}'
sage: f'{2^3}'
sage: x = 20
sage: f'{x} in binary is: {x:08b}'
'20 in binary is: 00010100'
sage: f'{list(map(lambda x: x^2, [1, 2, .., 5]))}'
'[1, 4, 9, 16, 25]'

Note that the format specifier is not preparsed. Expressions within it, however, are:

sage: f'{x:10r}'
Traceback (most recent call last):
ValueError: Unknown format code 'r' for object of type 'int'
sage: f'{x:{10r}}'
'        20'

Nested F-strings are also supported:

sage: f'{ f"{ 1/3 + 1/6 }" }'
sage: f'''1{ f"2{ f'4{ 2^3 }4' }2" }1'''


  • William Stein (2006-02-19): fixed bug when loading .py files

  • William Stein (2006-03-09): fixed crash in parsing exponentials; precision of real literals now determined by digits of input (like Mathematica)

  • Joe Wetherell (2006-04-14): added MAGMA-style constructor preparsing

  • Bobby Moretti (2007-01-25): added preliminary function assignment notation

  • Robert Bradshaw (2007-09-19): added strip_string_literals, containing_block utility functions. Arrr!; added [1,2,..,n] notation

  • Robert Bradshaw (2008-01-04): implicit multiplication (off by default)

  • Robert Bradshaw (2008-09-23): factor out constants

  • Robert Bradshaw (2009-01): simplify preparser by making it modular and using regular expressions; bug fixes, complex numbers, and binary input

class sage.repl.preparse.QuoteStack#

Bases: object

The preserved state of parsing in strip_string_literals().


Get the frame at the top of the stack or None if empty.


sage: qs = sage.repl.preparse.QuoteStack()
sage: qs.peek()
sage: qs.push(sage.repl.preparse.QuoteStackFrame('"'))
sage: qs.peek()

Remove and return the frame that was most recently added to the stack.

Raise an IndexError if the stack is empty.


sage: qs = sage.repl.preparse.QuoteStack()
sage: qs.pop()
Traceback (most recent call last):
IndexError: ...
sage: qs.push(sage.repl.preparse.QuoteStackFrame('"'))
sage: qs.pop()

Add a frame to the stack.

If the frame corresponds to an F-string, its delimiter is marked as no longer being a safe_delimiter().


sage: qs = sage.repl.preparse.QuoteStack()
sage: qs.push(sage.repl.preparse.QuoteStackFrame("'"))
sage: len(qs)

Return a string delimiter that may be safely inserted into the code output by strip_string_literals(), if any.

' is preferred over ". The triple-quoted versions are never returned since by the time they would be chosen, they would also be invalid. ''' cannot, for example, appear within an F-string delimited by '.

Once marked unsafe, a delimiter is never made safe again, even after the stack frame that used it is popped. It may no longer be applicable to parsing, but it appears somewhere in the processed code, so it is not safe to insert just anywhere. A future enhancement could be to map ranges in the processed code to the delimiter(s) that would be safe to insert there.


sage: from sage.repl.preparse import QuoteStack, QuoteStackFrame
sage: s = QuoteStack()
sage: s.safe_delimiter()
sage: s.push(QuoteStackFrame("'"))
sage: s.safe_delimiter()
sage: s.pop()
sage: s.push(QuoteStackFrame("'", f_string=True))
sage: s.safe_delimiter()
sage: s.push(QuoteStackFrame('"', f_string=True))
sage: s.safe_delimiter() is None
class sage.repl.preparse.QuoteStackFrame(delim, raw=False, f_string=False, braces=0, parens=0, brackets=0, fmt_spec=False, nested_fmt_spec=False)#

Bases: SimpleNamespace

The state of a single level of a string literal being parsed.

Only F-strings have more than one level.

sage.repl.preparse.containing_block(code, idx, delimiters=['()', '[]', '{}'], require_delim=True)#

Find the code block given by balanced delimiters that contains the position idx.


  • code - a string

  • idx - an integer; a starting position

  • delimiters - a list of strings (default: [‘()’, ‘[]’, ‘{}’]); the delimiters to balance. A delimiter must be a single character and no character can at the same time be opening and closing delimiter.

  • require_delim - a boolean (default: True); whether to raise a SyntaxError if delimiters are present. If the delimiters are unbalanced, an error will be raised in any case.


  • a 2-tuple (a,b) of integers, such that code[a:b] is delimited by balanced delimiters, a<=idx<b, and a is maximal and b is minimal with that property. If that does not exist, a SyntaxError is raised.

  • If require_delim is false and a,b as above can not be found, then 0, len(code) is returned.


sage: from sage.repl.preparse import containing_block
sage: s = "factor(next_prime(L[5]+1))"
sage: s[22]
sage: start, end = containing_block(s, 22)
sage: start, end
(17, 25)
sage: s[start:end]
sage: s[20]
sage: start, end = containing_block(s, 20); s[start:end]
sage: start, end = containing_block(s, 20, delimiters=['()']); s[start:end]
sage: start, end = containing_block(s, 10); s[start:end]

Pulls out numeric literals and assigns them to global variables. This eliminates the need to re-parse and create the literals, e.g., during every iteration of a loop.


  • code - a string; a block of code


  • a (string, string:string dictionary) 2-tuple; the block with literals replaced by variable names and a mapping from names to the new variables


sage: from sage.repl.preparse import extract_numeric_literals
sage: code, nums = extract_numeric_literals("1.2 + 5")
sage: print(code)
_sage_const_1p2  + _sage_const_5
sage: print(nums)
{'_sage_const_1p2': "RealNumber('1.2')", '_sage_const_5': 'Integer(5)'}

sage: extract_numeric_literals("[1, 1.1, 1e1, -1e-1, 1.]")[0]
'[_sage_const_1 , _sage_const_1p1 , _sage_const_1e1 , -_sage_const_1en1 , _sage_const_1p ]'

sage: extract_numeric_literals("[1.sqrt(), 1.2.sqrt(), 1r, 1.2r, R.1, R0.1, (1..5)]")[0]
'[_sage_const_1 .sqrt(), _sage_const_1p2 .sqrt(), 1 , 1.2 , R.1, R0.1, (_sage_const_1 .._sage_const_5 )]'
sage.repl.preparse.handle_encoding_declaration(contents, out)#

Find a PEP 263-style Python encoding declaration in the first or second line of contents. If found, output it to out and return contents without the encoding line; otherwise output a default UTF-8 declaration and return contents.


sage: from sage.repl.preparse import handle_encoding_declaration
sage: import sys
sage: c1='# -*- coding: latin-1 -*-\nimport os, sys\n...'
sage: c2='# -*- coding: iso-8859-15 -*-\nimport os, sys\n...'
sage: c3='# -*- coding: ascii -*-\nimport os, sys\n...'
sage: c4='import os, sys\n...'
sage: handle_encoding_declaration(c1, sys.stdout)
# -*- coding: latin-1 -*-
'import os, sys\n...'
sage: handle_encoding_declaration(c2, sys.stdout)
# -*- coding: iso-8859-15 -*-
'import os, sys\n...'
sage: handle_encoding_declaration(c3, sys.stdout)
# -*- coding: ascii -*-
'import os, sys\n...'
sage: handle_encoding_declaration(c4, sys.stdout)
# -*- coding: utf-8 -*-
'import os, sys\n...'


  • PEP 263 says that Python will interpret a UTF-8 byte order mark as a declaration of UTF-8 encoding, but I do not think we do that; this function only sees a Python string so it cannot account for a BOM.

  • We default to UTF-8 encoding even though PEP 263 says that Python files should default to ASCII.

  • Also see


sage.repl.preparse.implicit_mul(code, level=5)#

Insert *’s to make implicit multiplication explicit.



  • a string


sage: from sage.repl.preparse import implicit_mul
sage: implicit_mul('(2x^2-4x+3)a0')
sage: implicit_mul('a b c in L')
'a*b*c in L'
sage: implicit_mul('1r + 1e3 + 5exp(2)')
'1r + 1e3 + 5*exp(2)'
sage: implicit_mul('f(a)(b)', level=10)

Turn implicit multiplication on or off, optionally setting a specific level.


  • level – a boolean or integer (default: 5); how aggressive to be in placing *’s

    • 0 - Do nothing

    • 1 - Numeric followed by alphanumeric

    • 2 - Closing parentheses followed by alphanumeric

    • 3 - Spaces between alphanumeric

    • 10 - Adjacent parentheses (may mangle call statements)


The current level if no argument is given.


sage: implicit_multiplication(True)
sage: implicit_multiplication()
sage: preparse('2x')
sage: implicit_multiplication(False)
sage: preparse('2x')

Note that the IPython automagic feature cannot be used if level >= 3:

sage: implicit_multiplication(3)
sage: preparse('cd Documents')
sage: implicit_multiplication(2)
sage: preparse('cd Documents')
'cd Documents'
sage: implicit_multiplication(False)

In this case, one can use the explicit syntax for IPython magics such as %cd Documents.


Return True if s is a non-empty string of alphabetic characters or a non-empty string of digits or just a single _


sage: from sage.repl.preparse import isalphadigit_
sage: isalphadigit_('abc')
sage: isalphadigit_('123')
sage: isalphadigit_('_')
sage: isalphadigit_('a123')
sage.repl.preparse.parse_ellipsis(code, preparse_step=True)#

Preparses [0,2,..,n] notation.


  • code - a string

  • preparse_step - a boolean (default: True)


  • a string


sage: from sage.repl.preparse import parse_ellipsis
sage: parse_ellipsis("[1,2,..,n]")
sage: parse_ellipsis("for i in (f(x) .. L[10]):")
'for i in (ellipsis_iter(f(x) ,Ellipsis, L[10])):'
sage: [1.0..2.0]
[1.00000000000000, 2.00000000000000]
sage.repl.preparse.preparse(line, reset=True, do_time=False, ignore_prompts=False, numeric_literals=True)#

Preparse a line of input.


  • line - a string

  • reset - a boolean (default: True)

  • do_time - a boolean (default: False)

  • ignore_prompts - a boolean (default: False)

  • numeric_literals - a boolean (default: True)


  • a string


sage: preparse("ZZ.<x> = ZZ['x']")
"ZZ = ZZ['x']; (x,) = ZZ._first_ngens(1)"
sage: preparse("ZZ.<x> = ZZ['y']")
"ZZ = ZZ['y']; (x,) = ZZ._first_ngens(1)"
sage: preparse("ZZ.<x,y> = ZZ[]")
"ZZ = ZZ['x, y']; (x, y,) = ZZ._first_ngens(2)"
sage: preparse("ZZ.<x,y> = ZZ['u,v']")
"ZZ = ZZ['u,v']; (x, y,) = ZZ._first_ngens(2)"
sage: preparse("ZZ.<x> = QQ[2^(1/3)]")
'ZZ = QQ[Integer(2)**(Integer(1)/Integer(3))]; (x,) = ZZ._first_ngens(1)'
sage: QQ[2^(1/3)]                                                               # needs sage.rings.number_field sage.symbolic
Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?

sage: preparse("a^b")
sage: preparse("a^^b")
sage: 8^1
sage: 8^^1
sage: 9^^1

sage: preparse("A \\ B")
'A  * BackslashOperator() * B'
sage: preparse("A^2 \\ B + C")
'A**Integer(2)  * BackslashOperator() * B + C'
sage: preparse("a \\ b \\") # There is really only one backslash here, it is just being escaped.
'a  * BackslashOperator() * b \\'

sage: preparse("time R.<x> = ZZ[]", do_time=True)
'__time__ = cputime(); __wall__ = walltime(); R = ZZ[\'x\']; print("Time: CPU {:.2f} s, Wall: {:.2f} s".format(cputime(__time__), walltime(__wall__))); (x,) = R._first_ngens(1)'

Supports calculus-like function assignment, e.g., transforms:

f(x,y,z) = sin(x^3 - 4*y) + y^x


f = symbolic_expression(sin(x**3 - 4*y) + y**x).function(x,y,z)


  • Bobby Moretti

    • Initial version - 02/2007

  • William Stein

    • Make variables become defined if they are not already defined.

  • Robert Bradshaw

    • Rewrite using regular expressions (01/2009)


sage: preparse("f(x) = x^3-x")
'__tmp__=var("x"); f = symbolic_expression(x**Integer(3)-x).function(x)'
sage: preparse("f(u,v) = u - v")
'__tmp__=var("u,v"); f = symbolic_expression(u - v).function(u,v)'
sage: preparse("f(x) =-5")
'__tmp__=var("x"); f = symbolic_expression(-Integer(5)).function(x)'
sage: preparse("f(x) -= 5")
'f(x) -= Integer(5)'
sage: preparse("f(x_1, x_2) = x_1^2 - x_2^2")
'__tmp__=var("x_1,x_2"); f = symbolic_expression(x_1**Integer(2) - x_2**Integer(2)).function(x_1,x_2)'

For simplicity, this function assumes all statements begin and end with a semicolon:

sage: from sage.repl.preparse import preparse_calculus
sage: preparse_calculus(";f(t,s)=t^2;")
';__tmp__=var("t,s"); f = symbolic_expression(t^2).function(t,s);'
sage: preparse_calculus(";f( t , s ) = t^2;")
';__tmp__=var("t,s"); f = symbolic_expression(t^2).function(t,s);'
sage.repl.preparse.preparse_file(contents, globals=None, numeric_literals=True)#

Preparse contents which is input from a file such as .sage files.

Special attentions are given to numeric literals and load/attach file directives.


Temporarily, if @parallel is in the input, then numeric_literals is always set to False.


  • contents - a string

  • globals - dict or None (default: None); if given, then arguments to load/attach are evaluated in the namespace of this dict.

  • numeric_literals - bool (default: True), whether to factor out wrapping of integers and floats, so they do not get created repeatedly inside loops


  • a string


Preparse file named code{name} (presumably a .sage file), outputting to a temporary file. Returns name of temporary file.

sage.repl.preparse.preparse_file_named_to_stream(name, out)#

Preparse file named code{name} (presumably a .sage file), outputting to stream code{out}.


Parse generator syntax, converting:

obj.<gen0,gen1,...,genN> = objConstructor(...)


obj = objConstructor(..., names=("gen0", "gen1", ..., "genN"))
(gen0, gen1, ..., genN,) = obj.gens()


obj.<gen0,gen1,...,genN> = R[interior]


obj = R[interior]; (gen0, gen1, ..., genN,) = obj.gens()


  • code - a string


  • a string


  • The entire constructor must be on one line.


  • 2006-04-14: Joe Wetherell (

    • Initial version.

  • 2006-04-17: William Stein

    • Improvements to allow multiple statements.

  • 2006-05-01: William

    • Fix bug that Joe found.

  • 2006-10-31: William

    • Fix so obj does not have to be mutated.

  • 2009-01-27: Robert Bradshaw

    • Rewrite using regular expressions

sage.repl.preparse.preparse_numeric_literals(code, extract=False, quotes="'")#

Preparse numerical literals into their Sage counterparts, e.g. Integer, RealNumber, and ComplexNumber.


  • code - string; a code block to preparse

  • extract - boolean (default: False); whether to create names for the literals and return a dictionary of name-construction pairs

  • quotes - string (default: "'"); used to surround string arguments to RealNumber and ComplexNumber. If None, will rebuild the string using a list of its Unicode code-points.


  • a string or (string, string:string dictionary) 2-tuple; the preparsed block and, if extract is True, the name-construction mapping


sage: from sage.repl.preparse import preparse_numeric_literals
sage: preparse_numeric_literals("5")
sage: preparse_numeric_literals("5j")
"ComplexNumber(0, '5')"
sage: preparse_numeric_literals("5jr")
sage: preparse_numeric_literals("5l")
sage: preparse_numeric_literals("5L")
sage: preparse_numeric_literals("1.5")
sage: preparse_numeric_literals("1.5j")
"ComplexNumber(0, '1.5')"
sage: preparse_numeric_literals(".5j")
"ComplexNumber(0, '.5')"
sage: preparse_numeric_literals("5e9j")
"ComplexNumber(0, '5e9')"
sage: preparse_numeric_literals("5.")
sage: preparse_numeric_literals("5.j")
"ComplexNumber(0, '5.')"
sage: preparse_numeric_literals("")
sage: preparse_numeric_literals("")
sage: preparse_numeric_literals("")
"ComplexNumber(0, '5.5').foo()"
sage: preparse_numeric_literals("")
"ComplexNumber(0, '5').foo()"
sage: preparse_numeric_literals("1.exp()")
sage: preparse_numeric_literals("1e+10")
sage: preparse_numeric_literals("0x0af")
sage: preparse_numeric_literals("0x10.sqrt()")
sage: preparse_numeric_literals('0o100')
sage: preparse_numeric_literals('0b111001')
sage: preparse_numeric_literals('0xe')
sage: preparse_numeric_literals('0xEAR')
sage: preparse_numeric_literals('0x1012Fae')
sage: preparse_numeric_literals('042')
sage: preparse_numeric_literals('000042')

Test underscores as digit separators (PEP 515,

sage: preparse_numeric_literals('123_456')
sage: preparse_numeric_literals('123_456.78_9_0')
sage: preparse_numeric_literals('0b11_011')
sage: preparse_numeric_literals('0o76_321')
sage: preparse_numeric_literals('0xaa_aaa')
sage: preparse_numeric_literals('1_3.2_5e-2_2')

sage: for f in ["1_1.", "11_2.", "1.1_1", "1_1.1_1", ".1_1", ".1_1e1_1", ".1e1_1",
....:           "1e12_3", "1_1e1_1", "1.1_3e1_2", "1_1e1_1", "1e1", "1.e1_1",
....:           "1.0", "1_1.0"]:
....:     preparse_numeric_literals(f)
....:     assert preparse(f) == preparse_numeric_literals(f), f

Having consecutive underscores is not valid Python syntax, so it is not preparsed, and similarly with a trailing underscore:

sage: preparse_numeric_literals('123__45')
sage: 123__45
Traceback (most recent call last):
SyntaxError: invalid ...

sage: preparse_numeric_literals('3040_1_')
sage: 3040_1_
Traceback (most recent call last):
SyntaxError: invalid ...

Using the quotes parameter:

sage: preparse_numeric_literals('5j', quotes='"')
'ComplexNumber(0, "5")'
sage: preparse_numeric_literals('3.14', quotes="'''")
sage: preparse_numeric_literals('3.14', quotes=None)
'RealNumber(str().join(map(chr, [51, 46, 49, 52])))'
sage: preparse_numeric_literals('5j', quotes=None)
'ComplexNumber(0, str().join(map(chr, [53])))'

Remove leading sage: and >>> prompts so that pasting of examples from the documentation works.


  • line - a string to process


  • a string stripped of leading prompts


sage: from sage.repl.preparse import strip_prompts
sage: strip_prompts("sage: 2 + 2")
'2 + 2'
sage: strip_prompts(">>>   3 + 2")
'3 + 2'
sage: strip_prompts("  2 + 4")
'  2 + 4'
sage.repl.preparse.strip_string_literals(code, state=None)#

Return a string with all literal quotes replaced with labels and a dictionary of labels for re-substitution.

This makes parsing easier.


  • code - a string; the input

  • state - a QuoteStack (default: None); state with which to continue processing, e.g., across multiple calls to this function


  • a 3-tuple of the processed code, the dictionary of labels, and any accumulated state


sage: from sage.repl.preparse import strip_string_literals
sage: s, literals, state = strip_string_literals(r'''['a', "b", 'c', "d\""]''')
sage: s
'[%(L1)s, %(L2)s, %(L3)s, %(L4)s]'
sage: literals
{'L1': "'a'", 'L2': '"b"', 'L3': "'c'", 'L4': '"d\\""'}
sage: print(s % literals)
['a', "b", 'c', "d\""]
sage: print(strip_string_literals(r'-"\\\""-"\\"-')[0])

Triple-quotes are handled as well:

sage: s, literals, state = strip_string_literals("[a, '''b''', c, '']")
sage: s
'[a, %(L1)s, c, %(L2)s]'
sage: print(s % literals)
[a, '''b''', c, '']

Comments are substitute too:

sage: s, literals, state = strip_string_literals("code '#' # ccc 't'"); s
'code %(L1)s #%(L2)s'
sage: s % literals
"code '#' # ccc 't'"

A state is returned so one can break strings across multiple calls to this function:

sage: s, literals, state = strip_string_literals('s = "some'); s
's = %(L1)s'
sage: s, literals, state = strip_string_literals('thing" * 5', state); s
'%(L1)s * 5'