/* Math module -- standard C math library functions, pi and e */

/* Here are some comments from Tim Peters, extracted from the

   discussion attached to http://bugs.python.org/issue1640.  They

   describe the general aims of the math module with respect to

   special values, IEEE-754 floating-point exceptions, and Python

   exceptions.

These are the "spirit of 754" rules:

1. If the mathematical result is a real number, but of magnitude too

large to approximate by a machine float, overflow is signaled and the

result is an infinity (with the appropriate sign).

2. If the mathematical result is a real number, but of magnitude too

small to approximate by a machine float, underflow is signaled and the

result is a zero (with the appropriate sign).

3. At a singularity (a value x such that the limit of f(y) as y

approaches x exists and is an infinity), "divide by zero" is signaled

and the result is an infinity (with the appropriate sign).  This is

complicated a little by that the left-side and right-side limits may

not be the same; e.g., 1/x approaches +inf or -inf as x approaches 0

from the positive or negative directions.  In that specific case, the

sign of the zero determines the result of 1/0.

4. At a point where a function has no defined result in the extended

reals (i.e., the reals plus an infinity or two), invalid operation is

signaled and a NaN is returned.

And these are what Python has historically /tried/ to do (but not

always successfully, as platform libm behavior varies a lot):

For #1, raise OverflowError.

For #2, return a zero (with the appropriate sign if that happens by

accident ;-)).

For #3 and #4, raise ValueError.  It may have made sense to raise

Python's ZeroDivisionError in #3, but historically that's only been

raised for division by zero and mod by zero.

*/

/*

   In general, on an IEEE-754 platform the aim is to follow the C99

   standard, including Annex 'F', whenever possible.  Where the

   standard recommends raising the 'divide-by-zero' or 'invalid'

   floating-point exceptions, Python should raise a ValueError.  Where

   the standard recommends raising 'overflow', Python should raise an

   OverflowError.  In all other circumstances a value should be

   returned.

 */

#include "Python.h"

#include "longintrepr.h" /* just for SHIFT */

#ifdef _OSF_SOURCE

/* OSF1 5.1 doesn't make this available with XOPEN_SOURCE_EXTENDED defined */

extern double copysign(double, double);

#endif

/* Call is_error when errno != 0, and where x is the result libm

 * returned.  is_error will usually set up an exception and return

 * true (1), but may return false (0) without setting up an exception.

 */

static int

is_error(double x)

{

	int result = 1;	/* presumption of guilt */

	assert(errno);	/* non-zero errno is a precondition for calling */

	if (errno == EDOM)

		PyErr_SetString(PyExc_ValueError, "math domain error");

	else if (errno == ERANGE) {

		/* ANSI C generally requires libm functions to set ERANGE

		 * on overflow, but also generally *allows* them to set

		 * ERANGE on underflow too.  There's no consistency about

		 * the latter across platforms.

		 * Alas, C99 never requires that errno be set.

		 * Here we suppress the underflow errors (libm functions

		 * should return a zero on underflow, and +- HUGE_VAL on

		 * overflow, so testing the result for zero suffices to

		 * distinguish the cases).

		 *

		 * On some platforms (Ubuntu/ia64) it seems that errno can be

		 * set to ERANGE for subnormal results that do *not* underflow

		 * to zero.  So to be safe, we'll ignore ERANGE whenever the

		 * function result is less than one in absolute value.

		 */

		if (fabs(x) < 1.0)

			result = 0;

		else

			PyErr_SetString(PyExc_OverflowError,

					"math range error");

	}

	else

                /* Unexpected math error */

		PyErr_SetFromErrno(PyExc_ValueError);

	return result;

}

/*

   wrapper for atan2 that deals directly with special cases before

   delegating to the platform libm for the remaining cases.  This

   is necessary to get consistent behaviour across platforms.

   Windows, FreeBSD and alpha Tru64 are amongst platforms that don't

   always follow C99.

*/

static double

m_atan2(double y, double x)

{

	if (Py_IS_NAN(x) || Py_IS_NAN(y))

		return Py_NAN;

	if (Py_IS_INFINITY(y)) {

		if (Py_IS_INFINITY(x)) {

			if (copysign(1., x) == 1.)

				/* atan2(+-inf, +inf) == +-pi/4 */

				return copysign(0.25*Py_MATH_PI, y);

			else

				/* atan2(+-inf, -inf) == +-pi*3/4 */

				return copysign(0.75*Py_MATH_PI, y);

		}

		/* atan2(+-inf, x) == +-pi/2 for finite x */

		return copysign(0.5*Py_MATH_PI, y);

	}

	if (Py_IS_INFINITY(x) || y == 0.) {

		if (copysign(1., x) == 1.)

			/* atan2(+-y, +inf) = atan2(+-0, +x) = +-0. */

			return copysign(0., y);

		else

			/* atan2(+-y, -inf) = atan2(+-0., -x) = +-pi. */

			return copysign(Py_MATH_PI, y);

	}

	return atan2(y, x);

}

/*

   math_1 is used to wrap a libm function f that takes a double

   arguments and returns a double.

   The error reporting follows these rules, which are designed to do

   the right thing on C89/C99 platforms and IEEE 754/non IEEE 754

   platforms.

   - a NaN result from non-NaN inputs causes ValueError to be raised

   - an infinite result from finite inputs causes OverflowError to be

     raised if can_overflow is 1, or raises ValueError if can_overflow

     is 0.

   - if the result is finite and errno == EDOM then ValueError is

     raised

   - if the result is finite and nonzero and errno == ERANGE then

     OverflowError is raised

   The last rule is used to catch overflow on platforms which follow

   C89 but for which HUGE_VAL is not an infinity.

   For the majority of one-argument functions these rules are enough

   to ensure that Python's functions behave as specified in 'Annex F'

   of the C99 standard, with the 'invalid' and 'divide-by-zero'

   floating-point exceptions mapping to Python's ValueError and the

   'overflow' floating-point exception mapping to OverflowError.

   math_1 only works for functions that don't have singularities *and*

   the possibility of overflow; fortunately, that covers everything we

   care about right now.

*/

static PyObject *

math_1(PyObject *arg, double (*func) (double), int can_overflow)

{

	double x, r;

	x = PyFloat_AsDouble(arg);

	if (x == -1.0 && PyErr_Occurred())

		return NULL;

	errno = 0;

	PyFPE_START_PROTECT("in math_1", return 0);

	r = (*func)(x);

	PyFPE_END_PROTECT(r);

	if (Py_IS_NAN(r)) {

		if (!Py_IS_NAN(x))

			errno = EDOM;

		else

			errno = 0;

	}

	else if (Py_IS_INFINITY(r)) {

		if (Py_IS_FINITE(x))

			errno = can_overflow ? ERANGE : EDOM;

		else

			errno = 0;

	}

	if (errno && is_error(r))

		return NULL;

	else

		return PyFloat_FromDouble(r);

}

/*

   math_2 is used to wrap a libm function f that takes two double

   arguments and returns a double.

   The error reporting follows these rules, which are designed to do

   the right thing on C89/C99 platforms and IEEE 754/non IEEE 754

   platforms.

   - a NaN result from non-NaN inputs causes ValueError to be raised

   - an infinite result from finite inputs causes OverflowError to be

     raised.

   - if the result is finite and errno == EDOM then ValueError is

     raised

   - if the result is finite and nonzero and errno == ERANGE then

     OverflowError is raised

   The last rule is used to catch overflow on platforms which follow

   C89 but for which HUGE_VAL is not an infinity.

   For most two-argument functions (copysign, fmod, hypot, atan2)

   these rules are enough to ensure that Python's functions behave as

   specified in 'Annex F' of the C99 standard, with the 'invalid' and

   'divide-by-zero' floating-point exceptions mapping to Python's

   ValueError and the 'overflow' floating-point exception mapping to

   OverflowError.

*/

static PyObject *

math_2(PyObject *args, double (*func) (double, double), char *funcname)

{

	PyObject *ox, *oy;

	double x, y, r;

	if (! PyArg_UnpackTuple(args, funcname, 2, 2, &ox, &oy))

		return NULL;

	x = PyFloat_AsDouble(ox);

	y = PyFloat_AsDouble(oy);

	if ((x == -1.0 || y == -1.0) && PyErr_Occurred())

		return NULL;

	errno = 0;

	PyFPE_START_PROTECT("in math_2", return 0);

	r = (*func)(x, y);

	PyFPE_END_PROTECT(r);

	if (Py_IS_NAN(r)) {

		if (!Py_IS_NAN(x) && !Py_IS_NAN(y))

			errno = EDOM;

		else

			errno = 0;

	}

	else if (Py_IS_INFINITY(r)) {

		if (Py_IS_FINITE(x) && Py_IS_FINITE(y))

			errno = ERANGE;

		else

			errno = 0;

	}

	if (errno && is_error(r))

		return NULL;

	else

		return PyFloat_FromDouble(r);

}

#define FUNC1(funcname, func, can_overflow, docstring)			\

	static PyObject * math_##funcname(PyObject *self, PyObject *args) { \

		return math_1(args, func, can_overflow);		    \

	}\

        PyDoc_STRVAR(math_##funcname##_doc, docstring);

#define FUNC2(funcname, func, docstring) \

	static PyObject * math_##funcname(PyObject *self, PyObject *args) { \

		return math_2(args, func, #funcname); \

	}\

        PyDoc_STRVAR(math_##funcname##_doc, docstring);

FUNC1(acos, acos, 0,

      "acos(x)\n\nReturn the arc cosine (measured in radians) of x.")

FUNC1(acosh, acosh, 0,

      "acosh(x)\n\nReturn the hyperbolic arc cosine (measured in radians) of x.")

FUNC1(asin, asin, 0,

      "asin(x)\n\nReturn the arc sine (measured in radians) of x.")

FUNC1(asinh, asinh, 0,

      "asinh(x)\n\nReturn the hyperbolic arc sine (measured in radians) of x.")

FUNC1(atan, atan, 0,

      "atan(x)\n\nReturn the arc tangent (measured in radians) of x.")

FUNC2(atan2, m_atan2,

      "atan2(y, x)\n\nReturn the arc tangent (measured in radians) of y/x.\n"

      "Unlike atan(y/x), the signs of both x and y are considered.")

FUNC1(atanh, atanh, 0,

      "atanh(x)\n\nReturn the hyperbolic arc tangent (measured in radians) of x.")

FUNC1(ceil, ceil, 0,

      "ceil(x)\n\nReturn the ceiling of x as a float.\n"

      "This is the smallest integral value >= x.")

FUNC2(copysign, copysign,

      "copysign(x,y)\n\nReturn x with the sign of y.")

FUNC1(cos, cos, 0,

      "cos(x)\n\nReturn the cosine of x (measured in radians).")

FUNC1(cosh, cosh, 1,

      "cosh(x)\n\nReturn the hyperbolic cosine of x.")

FUNC1(exp, exp, 1,

      "exp(x)\n\nReturn e raised to the power of x.")

FUNC1(fabs, fabs, 0,

      "fabs(x)\n\nReturn the absolute value of the float x.")

FUNC1(floor, floor, 0,

      "floor(x)\n\nReturn the floor of x as a float.\n"

      "This is the largest integral value <= x.")

FUNC1(log1p, log1p, 1,

      "log1p(x)\n\nReturn the natural logarithm of 1+x (base e).\n\

      The result is computed in a way which is accurate for x near zero.")

FUNC1(sin, sin, 0,

      "sin(x)\n\nReturn the sine of x (measured in radians).")

FUNC1(sinh, sinh, 1,

      "sinh(x)\n\nReturn the hyperbolic sine of x.")

FUNC1(sqrt, sqrt, 0,

      "sqrt(x)\n\nReturn the square root of x.")

FUNC1(tan, tan, 0,

      "tan(x)\n\nReturn the tangent of x (measured in radians).")

FUNC1(tanh, tanh, 0,

      "tanh(x)\n\nReturn the hyperbolic tangent of x.")

/* Precision summation function as msum() by Raymond Hettinger in

   <http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/393090>,

   enhanced with the exact partials sum and roundoff from Mark

   Dickinson's post at <http://bugs.python.org/file10357/msum4.py>.

   See those links for more details, proofs and other references.

   Note 1: IEEE 754R floating point semantics are assumed,

   but the current implementation does not re-establish special

   value semantics across iterations (i.e. handling -Inf + Inf).

   Note 2:  No provision is made for intermediate overflow handling;

   therefore, sum([1e+308, 1e-308, 1e+308]) returns 1e+308 while

   sum([1e+308, 1e+308, 1e-308]) raises an OverflowError due to the

   overflow of the first partial sum.

   Note 3: The intermediate values lo, yr, and hi are declared volatile so

   aggressive compilers won't algebraically reduce lo to always be exactly 0.0.

   Also, the volatile declaration forces the values to be stored in memory as

   regular doubles instead of extended long precision (80-bit) values.  This

   prevents double rounding because any addition or subtraction of two doubles

   can be resolved exactly into double-sized hi and lo values.  As long as the 

   hi value gets forced into a double before yr and lo are computed, the extra

   bits in downstream extended precision operations (x87 for example) will be

   exactly zero and therefore can be losslessly stored back into a double,

   thereby preventing double rounding.

   Note 4: A similar implementation is in Modules/cmathmodule.c.

   Be sure to update both when making changes.

   Note 5: The signature of math.fsum() differs from __builtin__.sum()

   because the start argument doesn't make sense in the context of

   accurate summation.  Since the partials table is collapsed before

   returning a result, sum(seq2, start=sum(seq1)) may not equal the

   accurate result returned by sum(itertools.chain(seq1, seq2)).

*/

#define NUM_PARTIALS  32  /* initial partials array size, on stack */

/* Extend the partials array p[] by doubling its size. */

static int                          /* non-zero on error */

_fsum_realloc(double **p_ptr, Py_ssize_t  n,

             double  *ps,    Py_ssize_t *m_ptr)

{

	void *v = NULL;

	Py_ssize_t m = *m_ptr;

	m += m;  /* double */

	if (n < m && m < (PY_SSIZE_T_MAX / sizeof(double))) {

		double *p = *p_ptr;

		if (p == ps) {

			v = PyMem_Malloc(sizeof(double) * m);

			if (v != NULL)

				memcpy(v, ps, sizeof(double) * n);

		}

		else

			v = PyMem_Realloc(p, sizeof(double) * m);

	}

	if (v == NULL) {        /* size overflow or no memory */

		PyErr_SetString(PyExc_MemoryError, "math.fsum partials");

		return 1;

	}

	*p_ptr = (double*) v;

	*m_ptr = m;

	return 0;

}

/* Full precision summation of a sequence of floats.

   def msum(iterable):

       partials = []  # sorted, non-overlapping partial sums

       for x in iterable:

           i = 0

           for y in partials:

               if abs(x) < abs(y):

                   x, y = y, x

               hi = x + y

               lo = y - (hi - x)

               if lo:

                   partials[i] = lo

                   i += 1

               x = hi

           partials[i:] = [x]

       return sum_exact(partials)

   Rounded x+y stored in hi with the roundoff stored in lo.  Together hi+lo

   are exactly equal to x+y.  The inner loop applies hi/lo summation to each

   partial so that the list of partial sums remains exact.

   Sum_exact() adds the partial sums exactly and correctly rounds the final

   result (using the round-half-to-even rule).  The items in partials remain

   non-zero, non-special, non-overlapping and strictly increasing in

   magnitude, but possibly not all having the same sign.

   Depends on IEEE 754 arithmetic guarantees and half-even rounding.

*/

static PyObject*

math_fsum(PyObject *self, PyObject *seq)

{

	PyObject *item, *iter, *sum = NULL;

	Py_ssize_t i, j, n = 0, m = NUM_PARTIALS;

	double x, y, t, ps[NUM_PARTIALS], *p = ps;

	double xsave, special_sum = 0.0, inf_sum = 0.0;

	volatile double hi, yr, lo;

	iter = PyObject_GetIter(seq);

	if (iter == NULL)

		return NULL;

	PyFPE_START_PROTECT("fsum", Py_DECREF(iter); return NULL)

	for(;;) {           /* for x in iterable */

		assert(0 <= n && n <= m);

		assert((m == NUM_PARTIALS && p == ps) ||

		       (m >  NUM_PARTIALS && p != NULL));

		item = PyIter_Next(iter);

		if (item == NULL) {

			if (PyErr_Occurred())

				goto _fsum_error;

			break;

		}

		x = PyFloat_AsDouble(item);

		Py_DECREF(item);

		if (PyErr_Occurred())

			goto _fsum_error;

		xsave = x;

		for (i = j = 0; j < n; j++) {       /* for y in partials */

			y = p[j];

			if (fabs(x) < fabs(y)) {

				t = x; x = y; y = t;

			}

			hi = x + y;

			yr = hi - x;

			lo = y - yr;

			if (lo != 0.0)

				p[i++] = lo;

			x = hi;

		}

		n = i;                              /* ps[i:] = [x] */

		if (x != 0.0) {

			if (! Py_IS_FINITE(x)) {

				/* a nonfinite x could arise either as

				   a result of intermediate overflow, or

				   as a result of a nan or inf in the

				   summands */

				if (Py_IS_FINITE(xsave)) {

					PyErr_SetString(PyExc_OverflowError,

					      "intermediate overflow in fsum");

					goto _fsum_error;

				}

				if (Py_IS_INFINITY(xsave))

					inf_sum += xsave;

				special_sum += xsave;

				/* reset partials */

				n = 0;

			}

			else if (n >= m && _fsum_realloc(&p, n, ps, &m))

				goto _fsum_error;

			else

				p[n++] = x;

		}

	}

	if (special_sum != 0.0) {

		if (Py_IS_NAN(inf_sum))

			PyErr_SetString(PyExc_ValueError,

					"-inf + inf in fsum");

		else

			sum = PyFloat_FromDouble(special_sum);

		goto _fsum_error;

	}

	hi = 0.0;

	if (n > 0) {

		hi = p[--n];

		/* sum_exact(ps, hi) from the top, stop when the sum becomes

		   inexact. */

		while (n > 0) {

			x = hi;

			y = p[--n];

			assert(fabs(y) < fabs(x));

			hi = x + y;

			yr = hi - x;

			lo = y - yr;

			if (lo != 0.0)

				break;

		}

		/* Make half-even rounding work across multiple partials.

		   Needed so that sum([1e-16, 1, 1e16]) will round-up the last

		   digit to two instead of down to zero (the 1e-16 makes the 1

		   slightly closer to two).  With a potential 1 ULP rounding

		   error fixed-up, math.fsum() can guarantee commutativity. */

		if (n > 0 && ((lo < 0.0 && p[n-1] < 0.0) ||

			      (lo > 0.0 && p[n-1] > 0.0))) {

			y = lo * 2.0;

			x = hi + y;

			yr = x - hi;

			if (y == yr)

				hi = x;

		}

	}

	sum = PyFloat_FromDouble(hi);

_fsum_error:

	PyFPE_END_PROTECT(hi)

	Py_DECREF(iter);

	if (p != ps)

		PyMem_Free(p);

	return sum;

}

#undef NUM_PARTIALS