gh-38998: Allow construct RIF element from question-style string
    
As in the title. It allows you to say e.g. `RIF("1.23?2e-5")`.

Partially handles #36797. (only
for real case. Complex case is not handled yet, but in principle it
should not be too difficult.)

(I don't see any disadvantage of allowing this, it's backwards
compatible)

Issue: currently

```
sage: RIF("10", base=37)
37
sage: ZZ("10", base=37)
[error]
```

should this inconsistency be fixed? If so how?

[Edit: actually the rule of conversion should probably follow [string to
mpfr conversion rule](https://www.mpfr.org/mpfr-current/mpfr.html#index-
mpfr_005fstrtofr) or [string to mpz conversion
rule](https://gmplib.org/manual/Assigning-Integers) ]

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview. (there's no documentation change, but should we explicitly
mention the feature? I think the feature to construct from `[a..b]`
isn't explicitly mentioned either…?)

### ⌛ Dependencies

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#39001
    
URL: #38998
Reported by: user202729
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/libs/mpfi/__init__.pxd

@@ -26,7 +26,7 @@ cdef extern from "mpfi.h":
     int mpfi_set_z(mpfi_ptr, mpz_t)
     int mpfi_set_q(mpfi_ptr, mpq_t)
     int mpfi_set_fr(mpfi_ptr, mpfr_srcptr)
-    int mpfi_set_str(mpfi_ptr, char *, int)
+    int mpfi_set_str(mpfi_ptr, const char *, int)
 
     # combined initialization and assignment functions
     int mpfi_init_set(mpfi_ptr, mpfi_srcptr)
@@ -36,7 +36,7 @@ cdef extern from "mpfi.h":
     int mpfi_init_set_z(mpfi_ptr, mpz_srcptr)
     int mpfi_init_set_q(mpfi_ptr, mpq_srcptr)
     int mpfi_init_set_fr(mpfi_ptr, mpfr_srcptr)
-    int mpfi_init_set_str(mpfi_ptr, char *, int)
+    int mpfi_init_set_str(mpfi_ptr, const char *, int)
 
     # swapping two intervals
     void mpfi_swap(mpfi_ptr, mpfi_ptr)

src/sage/libs/mpfr/__init__.pxd

@@ -27,7 +27,7 @@ cdef extern from "mpfr.h":
     # int mpfr_set_f(mpfr_t rop, mpf_t op, mpfr_rnd_t rnd)
     int mpfr_set_ui_2exp(mpfr_t rop, unsigned long int op, mp_exp_t e, mpfr_rnd_t rnd)
     int mpfr_set_si_2exp(mpfr_t rop, long int op, mp_exp_t e, mpfr_rnd_t rnd)
-    int mpfr_set_str(mpfr_t rop,  char *s, int base, mpfr_rnd_t rnd)
+    int mpfr_set_str(mpfr_t rop,  const char *s, int base, mpfr_rnd_t rnd)
     int mpfr_strtofr(mpfr_t rop, char *nptr, char **endptr, int base, mpfr_rnd_t rnd)
     void mpfr_set_inf(mpfr_t x, int sign)
     void mpfr_set_nan(mpfr_t x)
@@ -43,7 +43,7 @@ cdef extern from "mpfr.h":
     int mpfr_init_set_z(mpfr_t rop, mpz_t op, mpfr_rnd_t rnd)
     int mpfr_init_set_q(mpfr_t rop, mpq_t op, mpfr_rnd_t rnd)
     # int mpfr_init_set_f(mpfr_t rop, mpf_t op, mpfr_rnd_t rnd)
-    int mpfr_init_set_str(mpfr_t x, char *s, int base, mpfr_rnd_t rnd)
+    int mpfr_init_set_str(mpfr_t x, const char *s, int base, mpfr_rnd_t rnd)
 
     # Conversion Functions
     double mpfr_get_d(mpfr_t op, mpfr_rnd_t rnd)

src/sage/rings/convert/mpfi.pyx

@@ -11,11 +11,16 @@ Convert Sage/Python objects to real/complex intervals
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
+import re
+
 from cpython.float cimport PyFloat_AS_DOUBLE
 from cpython.complex cimport PyComplex_RealAsDouble, PyComplex_ImagAsDouble
 
+from libc.stdio cimport printf
+
 from sage.libs.mpfr cimport *
 from sage.libs.mpfi cimport *
+from sage.libs.gmp.mpz cimport *
 from sage.libs.gsl.complex cimport *
 
 from sage.arith.long cimport integer_check_long
@@ -45,6 +50,243 @@ cdef inline int return_real(mpfi_ptr im) noexcept:
     return 0
 
 
+NUMBER = re.compile(rb'([+-]?(0[XxBb])?[0-9A-Za-z]+)\.([0-9A-Za-z]*)\?([0-9]*)(?:([EePp@])([+-]?[0-9]+))?')
+# example: -0xABC.DEF?12@5
+# match groups: (-0xABC) (0x) (DEF) (12) (@) (5)
+
+cdef int _from_str_question_style(mpfi_ptr x, bytes s, int base) except -1:
+    """
+    Convert a string in question style to an MPFI interval.
+
+    INPUT:
+
+    - ``x`` -- a pre-initialized MPFI interval
+
+    - ``s`` -- the string to convert
+
+    - ``base`` -- base to use for string conversion
+
+    OUTPUT:
+
+    - if conversion is possible: set ``x`` and return 0.
+
+    - in all other cases: return some nonzero value, or raise an exception.
+
+    TESTS:
+
+    Double check that ``ZZ``, ``RR`` and ``RIF`` follows the string
+    conversion rule for base different from `10` (except ``ZZ``
+    which only allows base up to `36`)::
+
+        sage: ZZ("0x123", base=0)
+        291
+        sage: RR("0x123.e1", base=0)  # rel tol 1e-12
+        291.878906250000
+        sage: RR("0x123.@1", base=0)  # rel tol 1e-12
+        4656.00000000000
+        sage: RIF("0x123.4@1", base=0)
+        4660
+        sage: ZZ("1Xx", base=36)  # case insensitive
+        2517
+        sage: ZZ("1Xx", base=62)
+        Traceback (most recent call last):
+        ...
+        ValueError: base (=62) must be 0 or between 2 and 36
+        sage: RR("1Xx", base=36)  # rel tol 1e-12
+        2517.00000000000
+        sage: RR("0x123", base=36)  # rel tol 1e-12
+        1.54101900000000e6
+        sage: RR("-1Xx@-1", base=62)  # rel tol 1e-12
+        -95.9516129032258
+        sage: RIF("1Xx@-1", base=62)  # rel tol 1e-12
+        95.95161290322580?
+        sage: RIF("1aE1", base=11)
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '1aE1' to real interval
+        sage: RIF("1aE1", base=11)
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '1aE1' to real interval
+
+    General checks::
+
+        sage: RIF("123456.?2").endpoints()  # rel tol 1e-12
+        (123454.0, 123458.0)
+        sage: RIF("1234.56?2").endpoints()  # rel tol 1e-12
+        (1234.54, 1234.58)
+        sage: RIF("1234.56?2e2").endpoints()  # rel tol 1e-12
+        (123454.0, 123458.0)
+        sage: x = RIF("-1234.56?2e2"); x.endpoints()  # rel tol 1e-12
+        (-123458.0, -123454.0)
+        sage: x
+        -1.2346?e5
+        sage: x.str(style="question", error_digits=1)
+        '-123456.?2'
+        sage: RIF("1.?100").endpoints()  # rel tol 1e-12
+        (-99.0, 101.0)
+        sage: RIF("1.?100").str(style="question", error_digits=3)
+        '1.?100'
+
+    Large exponent (ensure precision is not lost)::
+
+        sage: x = RIF("1.123456?2e100000000"); x
+        1.12346?e100000000
+        sage: x.str(style="question", error_digits=3)
+        '1.12345600?201e100000000'
+
+    Large precision::
+
+        sage: F = RealIntervalField(1000)
+        sage: x = F(sqrt(2)); x.endpoints()  # rel tol 1e-290
+        (1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798,
+         1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140799)
+        sage: x in F(x.str(style="question", error_digits=3))
+        True
+        sage: x in F(x.str(style="question", error_digits=0))
+        True
+        sage: F("1.123456789123456789123456789123456789123456789123456789123456789123456789?987654321987654321987654321e500").endpoints()  # rel tol 1e-290
+        (1.123456789123456789123456789123456789123456788135802467135802467135802468e500,
+         1.12345678912345678912345678912345678912345679011111111111111111111111111e500)
+
+    Stress test::
+
+        sage: for F in [RealIntervalField(15), RIF, RealIntervalField(100), RealIntervalField(1000)]:
+        ....:     for i in range(1000):
+        ....:         a, b = randint(-10^9, 10^9), randint(0, 50)
+        ....:         c, d = randint(-2^b, 2^b), randint(2, 5)
+        ....:         x = a * F(d)^c
+        ....:         assert x in F(x.str(style="question", error_digits=3)), (x, a, c, d)
+        ....:         assert x in F(x.str(style="question", error_digits=0)), (x, a, c, d)
+
+    Base different from `10` (note that the error and exponent are specified in decimal)::
+
+        sage: RIF("10000.?0", base=2).endpoints()  # rel tol 1e-12
+        (16.0, 16.0)
+        sage: RIF("10000.?0e10", base=2).endpoints()  # rel tol 1e-12
+        (16384.0, 16384.0)
+        sage: x = RIF("10000.?10", base=2); x.endpoints()  # rel tol 1e-12
+        (6.0, 26.0)
+        sage: x.str(base=2, style="question", error_digits=2)
+        '10000.000?80'
+        sage: x = RIF("10000.000?80", base=2); x.endpoints()  # rel tol 1e-12
+        (6.0, 26.0)
+        sage: x = RIF("12a.?", base=16); x.endpoints()  # rel tol 1e-12
+        (297.0, 299.0)
+        sage: x = RIF("12a.BcDeF?", base=16); x.endpoints()  # rel tol 1e-12
+        (298.737775802611, 298.737777709962)
+        sage: x = RIF("12a.BcDeF?@10", base=16); x.endpoints()  # rel tol 1e-12
+        (3.28465658150911e14, 3.28465660248065e14)
+        sage: x = RIF("12a.BcDeF?p10", base=16); x.endpoints()  # rel tol 1e-12
+        (305907.482421875, 305907.484375000)
+        sage: x = RIF("0x12a.BcDeF?p10", base=0); x.endpoints()  # rel tol 1e-12
+        (305907.482421875, 305907.484375000)
+
+    Space is allowed::
+
+        sage: RIF("-1234.56?2").endpoints()  # rel tol 1e-12
+        (-1234.58, -1234.54)
+        sage: RIF("- 1234.56 ?2").endpoints()  # rel tol 1e-12
+        (-1234.58, -1234.54)
+
+    Erroneous input::
+
+        sage: RIF("1234.56?2e2.3")
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '1234.56?2e2.3' to real interval
+        sage: RIF("1234?2")  # decimal point required
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '1234?2' to real interval
+        sage: RIF("1234.?2e")
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '1234.?2e' to real interval
+        sage: RIF("1.?e999999999999999999999999")
+        [-infinity .. +infinity]
+        sage: RIF("0X1.?", base=33)  # X is not valid digit in base 33
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '0X1.?' to real interval
+        sage: RIF("1.a?1e10", base=12)
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '1.a?1e10' to real interval
+        sage: RIF("1.1?a@10", base=12)
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '1.1?a@10' to real interval
+        sage: RIF("0x1?2e1", base=0)  # e is not allowed in base > 10, use @ instead
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '0x1?2e1' to real interval
+        sage: RIF("0x1?2p1", base=36)
+        Traceback (most recent call last):
+        ...
+        TypeError: unable to convert '0x1?2p1' to real interval
+    """
+    cdef mpz_t error_part
+    cdef mpfi_t error
+    cdef mpfr_t radius, neg_radius
+    cdef bytes int_part_string, base_prefix, frac_part_string, error_string, e, sci_expo_string, optional_expo, tmp
+
+    match = NUMBER.fullmatch(s)
+    if match is None:
+        return 1
+    int_part_string, base_prefix, frac_part_string, error_string, e, sci_expo_string = match.groups()
+
+    if (base > 10 or (base == 0 and base_prefix in (b'0X', b'0X'))) and e in (b'e', b'E'):
+        return 1
+    if base > 16 and e in (b'p', b'P'):
+        return 1
+    if base > 16 or not base_prefix:
+        base_prefix = b''
+
+    if error_string:
+        if mpz_init_set_str(error_part, error_string, 10):
+            mpz_clear(error_part)
+            return 1
+    else:
+        mpz_init_set_ui(error_part, 1)
+
+    optional_expo = e + sci_expo_string if e else b''
+    if mpfi_set_str(x, int_part_string + b'.' + frac_part_string + optional_expo, base):
+        mpz_clear(error_part)
+        return 1
+
+    mpfr_init2(radius, mpfi_get_prec(x))
+    tmp = base_prefix + (
+            b'0.' + b'0'*(len(frac_part_string)-1) + b'1' + optional_expo
+            if frac_part_string else
+            b'1.' + optional_expo)
+    # if base = 0:
+    #     when s = '-0x123.456@7', tmp = '0x0.001@7'
+    #     when s = '-0x123.@7', tmp = '0x1.@7'
+    # if base = 36:
+    #     when s = '-0x123.456@7', tmp = '0.001@7'
+    if mpfr_set_str(radius, tmp, base, MPFR_RNDU):
+        mpfr_clear(radius)
+        mpz_clear(error_part)
+        return 1
+
+    mpfr_mul_z(radius, radius, error_part, MPFR_RNDU)
+    mpz_clear(error_part)
+
+    mpfr_init2(neg_radius, mpfi_get_prec(x))
+    mpfr_neg(neg_radius, radius, MPFR_RNDD)
+
+    mpfi_init2(error, mpfi_get_prec(x))
+    mpfi_interv_fr(error, neg_radius, radius)
+    mpfr_clear(radius)
+    mpfr_clear(neg_radius)
+
+    mpfi_add(x, x, error)
+    mpfi_clear(error)
+
+    return 0
+
+
 cdef int mpfi_set_sage(mpfi_ptr re, mpfi_ptr im, x, field, int base) except -1:
     """
     Convert any object ``x`` to an MPFI interval or a pair of
@@ -186,6 +428,11 @@ cdef int mpfi_set_sage(mpfi_ptr re, mpfi_ptr im, x, field, int base) except -1:
         if isinstance(x, unicode):
             x = x.encode("ascii")
         if isinstance(x, bytes):
+            if b"?" in x:
+                if _from_str_question_style(re, (<bytes>x).replace(b' ', b''), base):
+                    x = bytes_to_str(x)
+                    raise TypeError(f"unable to convert {x!r} to real interval")
+                return return_real(im)
             s = (<bytes>x).replace(b'..', b',').replace(b' ', b'').replace(b'+infinity', b'@inf@').replace(b'-infinity', b'-@inf@')
             if mpfi_set_str(re, s, base):
                 x = bytes_to_str(x)