floatobject.c

/* Float object implementation */

/* XXX There should be overflow checks here, but it's hard to check
   for any kind of float exception without losing portability. */

#include "Python.h"
#include "pycore_abstract.h"      // _PyNumber_Index()
#include "pycore_dtoa.h"          // _Py_dg_dtoa()
#include "pycore_floatobject.h"   // _PyFloat_FormatAdvancedWriter()
#include "pycore_freelist.h"      // _Py_FREELIST_FREE(), _Py_FREELIST_POP()
#include "pycore_initconfig.h"    // _PyStatus_OK()
#include "pycore_long.h"          // _PyLong_GetOne()
#include "pycore_modsupport.h"    // _PyArg_NoKwnames()
#include "pycore_object.h"        // _PyObject_Init(), _PyDebugAllocatorStats()
#include "pycore_pymath.h"        // _PY_SHORT_FLOAT_REPR
#include "pycore_pystate.h"       // _PyInterpreterState_GET()
#include "pycore_stackref.h"      // PyStackRef_AsPyObjectBorrow()
#include "pycore_structseq.h"     // _PyStructSequence_FiniBuiltin()

#include <float.h>                // DBL_MAX
#include <stdlib.h>               // strtol()

/*[clinic input]
class float "PyObject *" "&PyFloat_Type"
[clinic start generated code]*/
/*[clinic end generated code: output=da39a3ee5e6b4b0d input=dd0003f68f144284]*/

#include "clinic/floatobject.c.h"


double
PyFloat_GetMax(void)
{
    return DBL_MAX;
}

double
PyFloat_GetMin(void)
{
    return DBL_MIN;
}

static PyTypeObject FloatInfoType;

PyDoc_STRVAR(floatinfo__doc__,
"sys.float_info\n\
\n\
A named tuple holding information about the float type. It contains low level\n\
information about the precision and internal representation. Please study\n\
your system's :file:`float.h` for more information.");

static PyStructSequence_Field floatinfo_fields[] = {
    {"max",             "DBL_MAX -- maximum representable finite float"},
    {"max_exp",         "DBL_MAX_EXP -- maximum int e such that radix**(e-1) "
                    "is representable"},
    {"max_10_exp",      "DBL_MAX_10_EXP -- maximum int e such that 10**e "
                    "is representable"},
    {"min",             "DBL_MIN -- Minimum positive normalized float"},
    {"min_exp",         "DBL_MIN_EXP -- minimum int e such that radix**(e-1) "
                    "is a normalized float"},
    {"min_10_exp",      "DBL_MIN_10_EXP -- minimum int e such that 10**e is "
                    "a normalized float"},
    {"dig",             "DBL_DIG -- maximum number of decimal digits that "
                    "can be faithfully represented in a float"},
    {"mant_dig",        "DBL_MANT_DIG -- mantissa digits"},
    {"epsilon",         "DBL_EPSILON -- Difference between 1 and the next "
                    "representable float"},
    {"radix",           "FLT_RADIX -- radix of exponent"},
    {"rounds",          "FLT_ROUNDS -- rounding mode used for arithmetic "
                    "operations"},
    {0}
};

static PyStructSequence_Desc floatinfo_desc = {
    "sys.float_info",           /* name */
    floatinfo__doc__,           /* doc */
    floatinfo_fields,           /* fields */
    11
};

PyObject *
PyFloat_GetInfo(void)
{
    PyObject* floatinfo;
    int pos = 0;

    floatinfo = PyStructSequence_New(&FloatInfoType);
    if (floatinfo == NULL) {
        return NULL;
    }

#define SetFlag(CALL) \
    do {                                                    \
        PyObject *flag = (CALL);                            \
        if (flag == NULL) {                                 \
            Py_CLEAR(floatinfo);                            \
            return NULL;                                    \
        }                                                   \
        PyStructSequence_SET_ITEM(floatinfo, pos++, flag);  \
    } while (0)

#define SetIntFlag(FLAG) SetFlag(PyLong_FromLong((FLAG)))
#define SetDblFlag(FLAG) SetFlag(PyFloat_FromDouble((FLAG)))

    SetDblFlag(DBL_MAX);
    SetIntFlag(DBL_MAX_EXP);
    SetIntFlag(DBL_MAX_10_EXP);
    SetDblFlag(DBL_MIN);
    SetIntFlag(DBL_MIN_EXP);
    SetIntFlag(DBL_MIN_10_EXP);
    SetIntFlag(DBL_DIG);
    SetIntFlag(DBL_MANT_DIG);
    SetDblFlag(DBL_EPSILON);
    SetIntFlag(FLT_RADIX);
    SetIntFlag(FLT_ROUNDS);
#undef SetIntFlag
#undef SetDblFlag
#undef SetFlag

    return floatinfo;
}

PyObject *
PyFloat_FromDouble(double fval)
{
    PyFloatObject *op = _Py_FREELIST_POP(PyFloatObject, floats);
    if (op == NULL) {
        op = PyObject_Malloc(sizeof(PyFloatObject));
        if (!op) {
            return PyErr_NoMemory();
        }
        _PyObject_Init((PyObject*)op, &PyFloat_Type);
    }
    op->ob_fval = fval;
    return (PyObject *) op;
}

_PyStackRef _PyFloat_FromDouble_ConsumeInputs(_PyStackRef left, _PyStackRef right, double value)
{
    PyStackRef_CLOSE_SPECIALIZED(left, _PyFloat_ExactDealloc);
    PyStackRef_CLOSE_SPECIALIZED(right, _PyFloat_ExactDealloc);
    return PyStackRef_FromPyObjectSteal(PyFloat_FromDouble(value));
}

static PyObject *
float_from_string_inner(const char *s, Py_ssize_t len, void *obj)
{
    double x;
    const char *end;
    const char *last = s + len;
    /* strip leading whitespace */
    while (s < last && Py_ISSPACE(*s)) {
        s++;
    }
    if (s == last) {
        PyErr_Format(PyExc_ValueError,
                     "could not convert string to float: "
                     "%R", obj);
        return NULL;
    }

    /* strip trailing whitespace */
    while (s < last - 1 && Py_ISSPACE(last[-1])) {
        last--;
    }

    /* We don't care about overflow or underflow.  If the platform
     * supports them, infinities and signed zeroes (on underflow) are
     * fine. */
    x = PyOS_string_to_double(s, (char **)&end, NULL);
    if (end != last) {
        PyErr_Format(PyExc_ValueError,
                     "could not convert string to float: "
                     "%R", obj);
        return NULL;
    }
    else if (x == -1.0 && PyErr_Occurred()) {
        return NULL;
    }
    else {
        return PyFloat_FromDouble(x);
    }
}

PyObject *
PyFloat_FromString(PyObject *v)
{
    const char *s;
    PyObject *s_buffer = NULL;
    Py_ssize_t len;
    Py_buffer view = {NULL, NULL};
    PyObject *result = NULL;

    if (PyUnicode_Check(v)) {
        s_buffer = _PyUnicode_TransformDecimalAndSpaceToASCII(v);
        if (s_buffer == NULL)
            return NULL;
        assert(PyUnicode_IS_ASCII(s_buffer));
        /* Simply get a pointer to existing ASCII characters. */
        s = PyUnicode_AsUTF8AndSize(s_buffer, &len);
        assert(s != NULL);
    }
    else if (PyBytes_Check(v)) {
        s = PyBytes_AS_STRING(v);
        len = PyBytes_GET_SIZE(v);
    }
    else if (PyByteArray_Check(v)) {
        s = PyByteArray_AS_STRING(v);
        len = PyByteArray_GET_SIZE(v);
    }
    else if (PyObject_GetBuffer(v, &view, PyBUF_SIMPLE) == 0) {
        s = (const char *)view.buf;
        len = view.len;
        /* Copy to NUL-terminated buffer. */
        s_buffer = PyBytes_FromStringAndSize(s, len);
        if (s_buffer == NULL) {
            PyBuffer_Release(&view);
            return NULL;
        }
        s = PyBytes_AS_STRING(s_buffer);
    }
    else {
        PyErr_Format(PyExc_TypeError,
            "float() argument must be a string or a real number, not '%.200s'",
            Py_TYPE(v)->tp_name);
        return NULL;
    }
    result = _Py_string_to_number_with_underscores(s, len, "float", v, v,
                                                   float_from_string_inner);
    PyBuffer_Release(&view);
    Py_XDECREF(s_buffer);
    return result;
}

void
_PyFloat_ExactDealloc(PyObject *obj)
{
    assert(PyFloat_CheckExact(obj));
    _Py_FREELIST_FREE(floats, obj, PyObject_Free);
}

static void
float_dealloc(PyObject *op)
{
    assert(PyFloat_Check(op));
    if (PyFloat_CheckExact(op))
        _PyFloat_ExactDealloc(op);
    else
        Py_TYPE(op)->tp_free(op);
}

double
PyFloat_AsDouble(PyObject *op)
{
    PyNumberMethods *nb;
    PyObject *res;
    double val;

    if (op == NULL) {
        PyErr_BadArgument();
        return -1;
    }

    if (PyFloat_Check(op)) {
        return PyFloat_AS_DOUBLE(op);
    }

    nb = Py_TYPE(op)->tp_as_number;
    if (nb == NULL || nb->nb_float == NULL) {
        if (nb && nb->nb_index) {
            PyObject *res = _PyNumber_Index(op);
            if (!res) {
                return -1;
            }
            double val = PyLong_AsDouble(res);
            Py_DECREF(res);
            return val;
        }
        PyErr_Format(PyExc_TypeError, "must be real number, not %.50s",
                     Py_TYPE(op)->tp_name);
        return -1;
    }

    res = (*nb->nb_float) (op);
    if (res == NULL) {
        return -1;
    }
    if (!PyFloat_CheckExact(res)) {
        if (!PyFloat_Check(res)) {
            PyErr_Format(PyExc_TypeError,
                         "%.50s.__float__ returned non-float (type %.50s)",
                         Py_TYPE(op)->tp_name, Py_TYPE(res)->tp_name);
            Py_DECREF(res);
            return -1;
        }
        if (PyErr_WarnFormat(PyExc_DeprecationWarning, 1,
                "%.50s.__float__ returned non-float (type %.50s).  "
                "The ability to return an instance of a strict subclass of float "
                "is deprecated, and may be removed in a future version of Python.",
                Py_TYPE(op)->tp_name, Py_TYPE(res)->tp_name)) {
            Py_DECREF(res);
            return -1;
        }
    }

    val = PyFloat_AS_DOUBLE(res);
    Py_DECREF(res);
    return val;
}

/* Macro and helper that convert PyObject obj to a C double and store
   the value in dbl.  If conversion to double raises an exception, obj is
   set to NULL, and the function invoking this macro returns NULL.  If
   obj is not of float or int type, Py_NotImplemented is incref'ed,
   stored in obj, and returned from the function invoking this macro.
*/
#define CONVERT_TO_DOUBLE(obj, dbl)                         \
    if (PyFloat_Check(obj))                                 \
        dbl = PyFloat_AS_DOUBLE(obj);                       \
    else if (_Py_convert_int_to_double(&(obj), &(dbl)) < 0) \
        return obj;

/* Methods */

int
_Py_convert_int_to_double(PyObject **v, double *dbl)
{
    PyObject *obj = *v;

    if (PyLong_Check(obj)) {
        *dbl = PyLong_AsDouble(obj);
        if (*dbl == -1.0 && PyErr_Occurred()) {
            *v = NULL;
            return -1;
        }
    }
    else {
        *v = Py_NewRef(Py_NotImplemented);
        return -1;
    }
    return 0;
}

static PyObject *
float_repr(PyObject *op)
{
    PyFloatObject *v = _PyFloat_CAST(op);
    PyObject *result;
    char *buf;

    buf = PyOS_double_to_string(PyFloat_AS_DOUBLE(v),
                                'r', 0,
                                Py_DTSF_ADD_DOT_0,
                                NULL);
    if (!buf)
        return PyErr_NoMemory();
    result = _PyUnicode_FromASCII(buf, strlen(buf));
    PyMem_Free(buf);
    return result;
}

/* Comparison is pretty much a nightmare.  When comparing float to float,
 * we do it as straightforwardly (and long-windedly) as conceivable, so
 * that, e.g., Python x == y delivers the same result as the platform
 * C x == y when x and/or y is a NaN.
 * When mixing float with an integer type, there's no good *uniform* approach.
 * Converting the double to an integer obviously doesn't work, since we
 * may lose info from fractional bits.  Converting the integer to a double
 * also has two failure modes:  (1) an int may trigger overflow (too
 * large to fit in the dynamic range of a C double); (2) even a C long may have
 * more bits than fit in a C double (e.g., on a 64-bit box long may have
 * 63 bits of precision, but a C double probably has only 53), and then
 * we can falsely claim equality when low-order integer bits are lost by
 * coercion to double.  So this part is painful too.
 */

static PyObject*
float_richcompare(PyObject *v, PyObject *w, int op)
{
    double i, j;
    int r = 0;

    assert(PyFloat_Check(v));
    i = PyFloat_AS_DOUBLE(v);

    /* Switch on the type of w.  Set i and j to doubles to be compared,
     * and op to the richcomp to use.
     */
    if (PyFloat_Check(w))
        j = PyFloat_AS_DOUBLE(w);

    else if (!isfinite(i)) {
        if (PyLong_Check(w))
            /* If i is an infinity, its magnitude exceeds any
             * finite integer, so it doesn't matter which int we
             * compare i with.  If i is a NaN, similarly.
             */
            j = 0.0;
        else
            goto Unimplemented;
    }

    else if (PyLong_Check(w)) {
        int vsign = i == 0.0 ? 0 : i < 0.0 ? -1 : 1;
        int wsign;
        int exponent;

        (void)PyLong_GetSign(w, &wsign);
        if (vsign != wsign) {
            /* Magnitudes are irrelevant -- the signs alone
             * determine the outcome.
             */
            i = (double)vsign;
            j = (double)wsign;
            goto Compare;
        }
        /* The signs are the same. */
        /* Convert w to a double if it fits.  In particular, 0 fits. */
        int64_t nbits64 = _PyLong_NumBits(w);
        assert(nbits64 >= 0);
        assert(!PyErr_Occurred());
        if (nbits64 > DBL_MAX_EXP) {
            /* This Python integer is larger than any finite C double.
             * Replace with little doubles
             * that give the same outcome -- w is so large that
             * its magnitude must exceed the magnitude of any
             * finite float.
             */
            i = (double)vsign;
            assert(wsign != 0);
            j = wsign * 2.0;
            goto Compare;
        }
        int nbits = (int)nbits64;
        if (nbits <= 48) {
            j = PyLong_AsDouble(w);
            /* It's impossible that <= 48 bits overflowed. */
            assert(j != -1.0 || ! PyErr_Occurred());
            goto Compare;
        }
        assert(wsign != 0); /* else nbits was 0 */
        assert(vsign != 0); /* if vsign were 0, then since wsign is
                             * not 0, we would have taken the
                             * vsign != wsign branch at the start */
        /* We want to work with non-negative numbers. */
        if (vsign < 0) {
            /* "Multiply both sides" by -1; this also swaps the
             * comparator.
             */
            i = -i;
            op = _Py_SwappedOp[op];
        }
        assert(i > 0.0);
        (void) frexp(i, &exponent);
        /* exponent is the # of bits in v before the radix point;
         * we know that nbits (the # of bits in w) > 48 at this point
         */
        if (exponent < nbits) {
            i = 1.0;
            j = 2.0;
            goto Compare;
        }
        if (exponent > nbits) {
            i = 2.0;
            j = 1.0;
            goto Compare;
        }
        /* v and w have the same number of bits before the radix
         * point.  Construct two ints that have the same comparison
         * outcome.
         */
        {
            double fracpart;
            double intpart;
            PyObject *result = NULL;
            PyObject *vv = NULL;
            PyObject *ww = w;

            if (wsign < 0) {
                ww = PyNumber_Negative(w);
                if (ww == NULL)
                    goto Error;
            }
            else
                Py_INCREF(ww);

            fracpart = modf(i, &intpart);
            vv = PyLong_FromDouble(intpart);
            if (vv == NULL)
                goto Error;

            if (fracpart != 0.0) {
                /* Shift left, and or a 1 bit into vv
                 * to represent the lost fraction.
                 */
                PyObject *temp;

                temp = _PyLong_Lshift(ww, 1);
                if (temp == NULL)
                    goto Error;
                Py_SETREF(ww, temp);

                temp = _PyLong_Lshift(vv, 1);
                if (temp == NULL)
                    goto Error;
                Py_SETREF(vv, temp);

                temp = PyNumber_Or(vv, _PyLong_GetOne());
                if (temp == NULL)
                    goto Error;
                Py_SETREF(vv, temp);
            }

            r = PyObject_RichCompareBool(vv, ww, op);
            if (r < 0)
                goto Error;
            result = PyBool_FromLong(r);
         Error:
            Py_XDECREF(vv);
            Py_XDECREF(ww);
            return result;
        }
    } /* else if (PyLong_Check(w)) */

    else        /* w isn't float or int */
        goto Unimplemented;

 Compare:
    switch (op) {
    case Py_EQ:
        r = i == j;
        break;
    case Py_NE:
        r = i != j;
        break;
    case Py_LE:
        r = i <= j;
        break;
    case Py_GE:
        r = i >= j;
        break;
    case Py_LT:
        r = i < j;
        break;
    case Py_GT:
        r = i > j;
        break;
    }
    return PyBool_FromLong(r);

 Unimplemented:
    Py_RETURN_NOTIMPLEMENTED;
}

static Py_hash_t
float_hash(PyObject *op)
{
    PyFloatObject *v = _PyFloat_CAST(op);
    return _Py_HashDouble(op, v->ob_fval);
}

static PyObject *
float_add(PyObject *v, PyObject *w)
{
    double a,b;
    CONVERT_TO_DOUBLE(v, a);
    CONVERT_TO_DOUBLE(w, b);
    a = a + b;
    return PyFloat_FromDouble(a);
}

static PyObject *
float_sub(PyObject *v, PyObject *w)
{
    double a,b;
    CONVERT_TO_DOUBLE(v, a);
    CONVERT_TO_DOUBLE(w, b);
    a = a - b;
    return PyFloat_FromDouble(a);
}

static PyObject *
float_mul(PyObject *v, PyObject *w)
{
    double a,b;
    CONVERT_TO_DOUBLE(v, a);
    CONVERT_TO_DOUBLE(w, b);
    a = a * b;
    return PyFloat_FromDouble(a);
}

static PyObject *
float_div(PyObject *v, PyObject *w)
{
    double a,b;
    CONVERT_TO_DOUBLE(v, a);
    CONVERT_TO_DOUBLE(w, b);
    if (b == 0.0) {
        PyErr_SetString(PyExc_ZeroDivisionError,
                        "division by zero");
        return NULL;
    }
    a = a / b;
    return PyFloat_FromDouble(a);
}

static PyObject *
float_rem(PyObject *v, PyObject *w)
{
    double vx, wx;
    double mod;
    CONVERT_TO_DOUBLE(v, vx);
    CONVERT_TO_DOUBLE(w, wx);
    if (wx == 0.0) {
        PyErr_SetString(PyExc_ZeroDivisionError,
                        "division by zero");
        return NULL;
    }
    mod = fmod(vx, wx);
    if (mod) {
        /* ensure the remainder has the same sign as the denominator */
        if ((wx < 0) != (mod < 0)) {
            mod += wx;
        }
    }
    else {
        /* the remainder is zero, and in the presence of signed zeroes
           fmod returns different results across platforms; ensure
           it has the same sign as the denominator. */
        mod = copysign(0.0, wx);
    }
    return PyFloat_FromDouble(mod);
}

static void
_float_div_mod(double vx, double wx, double *floordiv, double *mod)
{
    double div;
    *mod = fmod(vx, wx);
    /* fmod is typically exact, so vx-mod is *mathematically* an
       exact multiple of wx.  But this is fp arithmetic, and fp
       vx - mod is an approximation; the result is that div may
       not be an exact integral value after the division, although
       it will always be very close to one.
    */
    div = (vx - *mod) / wx;
    if (*mod) {
        /* ensure the remainder has the same sign as the denominator */
        if ((wx < 0) != (*mod < 0)) {
            *mod += wx;
            div -= 1.0;
        }
    }
    else {
        /* the remainder is zero, and in the presence of signed zeroes
           fmod returns different results across platforms; ensure
           it has the same sign as the denominator. */
        *mod = copysign(0.0, wx);
    }
    /* snap quotient to nearest integral value */
    if (div) {
        *floordiv = floor(div);
        if (div - *floordiv > 0.5) {
            *floordiv += 1.0;
        }
    }
    else {
        /* div is zero - get the same sign as the true quotient */
        *floordiv = copysign(0.0, vx / wx); /* zero w/ sign of vx/wx */
    }
}

static PyObject *
float_divmod(PyObject *v, PyObject *w)
{
    double vx, wx;
    double mod, floordiv;
    CONVERT_TO_DOUBLE(v, vx);
    CONVERT_TO_DOUBLE(w, wx);
    if (wx == 0.0) {
        PyErr_SetString(PyExc_ZeroDivisionError, "division by zero");
        return NULL;
    }
    _float_div_mod(vx, wx, &floordiv, &mod);
    return Py_BuildValue("(dd)", floordiv, mod);
}

static PyObject *
float_floor_div(PyObject *v, PyObject *w)
{
    double vx, wx;
    double mod, floordiv;
    CONVERT_TO_DOUBLE(v, vx);
    CONVERT_TO_DOUBLE(w, wx);
    if (wx == 0.0) {
        PyErr_SetString(PyExc_ZeroDivisionError, "division by zero");
        return NULL;
    }
    _float_div_mod(vx, wx, &floordiv, &mod);
    return PyFloat_FromDouble(floordiv);
}

/* determine whether x is an odd integer or not;  assumes that
   x is not an infinity or nan. */
#define DOUBLE_IS_ODD_INTEGER(x) (fmod(fabs(x), 2.0) == 1.0)

static PyObject *
float_pow(PyObject *v, PyObject *w, PyObject *z)
{
    double iv, iw, ix;
    int negate_result = 0;

    if ((PyObject *)z != Py_None) {
        PyErr_SetString(PyExc_TypeError, "pow() 3rd argument not "
            "allowed unless all arguments are integers");
        return NULL;
    }

    CONVERT_TO_DOUBLE(v, iv);
    CONVERT_TO_DOUBLE(w, iw);

    /* Sort out special cases here instead of relying on pow() */
    if (iw == 0) {              /* v**0 is 1, even 0**0 */
        return PyFloat_FromDouble(1.0);
    }
    if (isnan(iv)) {        /* nan**w = nan, unless w == 0 */
        return PyFloat_FromDouble(iv);
    }
    if (isnan(iw)) {        /* v**nan = nan, unless v == 1; 1**nan = 1 */
        return PyFloat_FromDouble(iv == 1.0 ? 1.0 : iw);
    }
    if (isinf(iw)) {
        /* v**inf is: 0.0 if abs(v) < 1; 1.0 if abs(v) == 1; inf if
         *     abs(v) > 1 (including case where v infinite)
         *
         * v**-inf is: inf if abs(v) < 1; 1.0 if abs(v) == 1; 0.0 if
         *     abs(v) > 1 (including case where v infinite)
         */
        iv = fabs(iv);
        if (iv == 1.0)
            return PyFloat_FromDouble(1.0);
        else if ((iw > 0.0) == (iv > 1.0))
            return PyFloat_FromDouble(fabs(iw)); /* return inf */
        else
            return PyFloat_FromDouble(0.0);
    }
    if (isinf(iv)) {
        /* (+-inf)**w is: inf for w positive, 0 for w negative; in
         *     both cases, we need to add the appropriate sign if w is
         *     an odd integer.
         */
        int iw_is_odd = DOUBLE_IS_ODD_INTEGER(iw);
        if (iw > 0.0)
            return PyFloat_FromDouble(iw_is_odd ? iv : fabs(iv));
        else
            return PyFloat_FromDouble(iw_is_odd ?
                                      copysign(0.0, iv) : 0.0);
    }
    if (iv == 0.0) {  /* 0**w is: 0 for w positive, 1 for w zero
                         (already dealt with above), and an error
                         if w is negative. */
        int iw_is_odd = DOUBLE_IS_ODD_INTEGER(iw);
        if (iw < 0.0) {
            PyErr_SetString(PyExc_ZeroDivisionError,
                            "zero to a negative power");
            return NULL;
        }
        /* use correct sign if iw is odd */
        return PyFloat_FromDouble(iw_is_odd ? iv : 0.0);
    }

    if (iv < 0.0) {
        /* Whether this is an error is a mess, and bumps into libm
         * bugs so we have to figure it out ourselves.
         */
        if (iw != floor(iw)) {
            /* Negative numbers raised to fractional powers
             * become complex.
             */
            return PyComplex_Type.tp_as_number->nb_power(v, w, z);
        }
        /* iw is an exact integer, albeit perhaps a very large
         * one.  Replace iv by its absolute value and remember
         * to negate the pow result if iw is odd.
         */
        iv = -iv;
        negate_result = DOUBLE_IS_ODD_INTEGER(iw);
    }

    if (iv == 1.0) { /* 1**w is 1, even 1**inf and 1**nan */
        /* (-1) ** large_integer also ends up here.  Here's an
         * extract from the comments for the previous
         * implementation explaining why this special case is
         * necessary:
         *
         * -1 raised to an exact integer should never be exceptional.
         * Alas, some libms (chiefly glibc as of early 2003) return
         * NaN and set EDOM on pow(-1, large_int) if the int doesn't
         * happen to be representable in a *C* integer.  That's a
         * bug.
         */
        return PyFloat_FromDouble(negate_result ? -1.0 : 1.0);
    }

    /* Now iv and iw are finite, iw is nonzero, and iv is
     * positive and not equal to 1.0.  We finally allow
     * the platform pow to step in and do the rest.
     */
    errno = 0;
    ix = pow(iv, iw);
    _Py_ADJUST_ERANGE1(ix);
    if (negate_result)
        ix = -ix;

    if (errno != 0) {
        /* We don't expect any errno value other than ERANGE, but
         * the range of libm bugs appears unbounded.
         */
        PyErr_SetFromErrno(errno == ERANGE ? PyExc_OverflowError :
                             PyExc_ValueError);
        return NULL;
    }
    return PyFloat_FromDouble(ix);
}

#undef DOUBLE_IS_ODD_INTEGER

static PyObject *
float_neg(PyObject *op)
{
    PyFloatObject *v = _PyFloat_CAST(op);
    return PyFloat_FromDouble(-v->ob_fval);
}

static PyObject *
float_abs(PyObject *op)
{
    PyFloatObject *v = _PyFloat_CAST(op);
    return PyFloat_FromDouble(fabs(v->ob_fval));
}

static int
float_bool(PyObject *op)
{
    PyFloatObject *v = _PyFloat_CAST(op);
    return v->ob_fval != 0.0;
}

/*[clinic input]
float.is_integer

Return True if the float is an integer.
[clinic start generated code]*/

static PyObject *
float_is_integer_impl(PyObject *self)
/*[clinic end generated code: output=7112acf95a4d31ea input=311810d3f777e10d]*/
{
    double x = PyFloat_AsDouble(self);
    PyObject *o;

    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    if (!isfinite(x))
        Py_RETURN_FALSE;
    errno = 0;
    o = (floor(x) == x) ? Py_True : Py_False;
    if (errno != 0) {
        PyErr_SetFromErrno(errno == ERANGE ? PyExc_OverflowError :
                             PyExc_ValueError);
        return NULL;
    }
    return Py_NewRef(o);
}

/*[clinic input]
float.__trunc__

Return the Integral closest to x between 0 and x.
[clinic start generated code]*/

static PyObject *
float___trunc___impl(PyObject *self)
/*[clinic end generated code: output=dd3e289dd4c6b538 input=591b9ba0d650fdff]*/
{
    return PyLong_FromDouble(PyFloat_AS_DOUBLE(self));
}

/*[clinic input]
float.__floor__

Return the floor as an Integral.
[clinic start generated code]*/

static PyObject *
float___floor___impl(PyObject *self)
/*[clinic end generated code: output=e0551dbaea8c01d1 input=77bb13eb12e268df]*/
{
    double x = PyFloat_AS_DOUBLE(self);
    return PyLong_FromDouble(floor(x));
}

/*[clinic input]
float.__ceil__

Return the ceiling as an Integral.
[clinic start generated code]*/

static PyObject *
float___ceil___impl(PyObject *self)
/*[clinic end generated code: output=a2fd8858f73736f9 input=79e41ae94aa0a516]*/
{
    double x = PyFloat_AS_DOUBLE(self);
    return PyLong_FromDouble(ceil(x));
}

/* double_round: rounds a finite double to the closest multiple of
   10**-ndigits; here ndigits is within reasonable bounds (typically, -308 <=
   ndigits <= 323).  Returns a Python float, or sets a Python error and
   returns NULL on failure (OverflowError and memory errors are possible). */

#if _PY_SHORT_FLOAT_REPR == 1
/* version of double_round that uses the correctly-rounded string<->double
   conversions from Python/dtoa.c */

static PyObject *
double_round(double x, int ndigits) {

    double rounded;
    Py_ssize_t buflen, mybuflen=100;
    char *buf, *buf_end, shortbuf[100], *mybuf=shortbuf;
    int decpt, sign;
    PyObject *result = NULL;
    _Py_SET_53BIT_PRECISION_HEADER;

    /* round to a decimal string */
    _Py_SET_53BIT_PRECISION_START;
    buf = _Py_dg_dtoa(x, 3, ndigits, &decpt, &sign, &buf_end);
    _Py_SET_53BIT_PRECISION_END;
    if (buf == NULL) {
        PyErr_NoMemory();
        return NULL;
    }

    /* Get new buffer if shortbuf is too small.  Space needed <= buf_end -
    buf + 8: (1 extra for '0', 1 for sign, 5 for exp, 1 for '\0').  */
    buflen = buf_end - buf;
    if (buflen + 8 > mybuflen) {
        mybuflen = buflen+8;
        mybuf = (char *)PyMem_Malloc(mybuflen);
        if (mybuf == NULL) {
            PyErr_NoMemory();
            goto exit;
        }
    }
    /* copy buf to mybuf, adding exponent, sign and leading 0 */
    PyOS_snprintf(mybuf, mybuflen, "%s0%se%d", (sign ? "-" : ""),
                  buf, decpt - (int)buflen);

    /* and convert the resulting string back to a double */
    errno = 0;
    _Py_SET_53BIT_PRECISION_START;
    rounded = _Py_dg_strtod(mybuf, NULL);
    _Py_SET_53BIT_PRECISION_END;
    if (errno == ERANGE && fabs(rounded) >= 1.)
        PyErr_SetString(PyExc_OverflowError,
                        "rounded value too large to represent");
    else
        result = PyFloat_FromDouble(rounded);

    /* done computing value;  now clean up */
    if (mybuf != shortbuf)
        PyMem_Free(mybuf);
  exit:
    _Py_dg_freedtoa(buf);
    return result;
}

#else  // _PY_SHORT_FLOAT_REPR == 0

/* fallback version, to be used when correctly rounded binary<->decimal
   conversions aren't available */

static PyObject *
double_round(double x, int ndigits) {
    double pow1, pow2, y, z;
    if (ndigits >= 0) {
        if (ndigits > 22) {
            /* pow1 and pow2 are each safe from overflow, but
               pow1*pow2 ~= pow(10.0, ndigits) might overflow */
            pow1 = pow(10.0, (double)(ndigits-22));
            pow2 = 1e22;
        }
        else {
            pow1 = pow(10.0, (double)ndigits);
            pow2 = 1.0;
        }
        y = (x*pow1)*pow2;
        /* if y overflows, then rounded value is exactly x */
        if (!isfinite(y))