I finally got the time to update and merge Mark's and my trunk-math branch. The patch is collaborated work of Mark Dickinson and me. It was mostly done a few months ago. The patch fixes a lot of loose ends and edge cases related to operations with NaN, INF, very small values and complex math.
The patch also adds acosh, asinh, atanh, log1p and copysign to all platforms. Finally it fixes differences between platforms like different results or exceptions for edge cases. Have fun :)
diff --git a/Modules/mathmodule.c b/Modules/mathmodule.c
index 63a2cca..56b52fd 100644
--- a/Modules/mathmodule.c
+++ b/Modules/mathmodule.c
@@ -1,17 +1,60 @@
/* 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 */
-#ifndef _MSC_VER
-#ifndef __STDC__
-extern double fmod (double, double);
-extern double frexp (double, int *);
-extern double ldexp (double, int);
-extern double modf (double, double *);
-#endif /* __STDC__ */
-#endif /* _MSC_VER */
-
#ifdef _OSF_SOURCE
/* OSF1 5.1 doesn't make this available with XOPEN_SOURCE_EXTENDED defined */
extern double copysign(double, double);
@@ -52,28 +95,97 @@
return result;
}
+/*
+ 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))
+math_1(PyObject *arg, double (*func) (double), int can_overflow)
{
- double x = PyFloat_AsDouble(arg);
+ 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)
- x = (*func)(x);
- PyFPE_END_PROTECT(x)
- Py_SET_ERRNO_ON_MATH_ERROR(x);
- if (errno && is_error(x))
+ 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(x);
+ 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;
+ double x, y, r;
if (! PyArg_UnpackTuple(args, funcname, 2, 2, &ox, &oy))
return NULL;
x = PyFloat_AsDouble(ox);
@@ -81,19 +193,30 @@
if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
return NULL;
errno = 0;
- PyFPE_START_PROTECT("in math_2", return 0)
- x = (*func)(x, y);
- PyFPE_END_PROTECT(x)
- Py_SET_ERRNO_ON_MATH_ERROR(x);
- if (errno && is_error(x))
+ 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(x);
+ return PyFloat_FromDouble(r);
}
-#define FUNC1(funcname, func, docstring) \
+#define FUNC1(funcname, func, can_overflow, docstring) \
static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
- return math_1(args, func); \
+ return math_1(args, func, can_overflow); \
}\
PyDoc_STRVAR(math_##funcname##_doc, docstring);
@@ -103,55 +226,49 @@
}\
PyDoc_STRVAR(math_##funcname##_doc, docstring);
-FUNC1(acos, acos,
+FUNC1(acos, acos, 0,
"acos(x)\n\nReturn the arc cosine (measured in radians) of x.")
-FUNC1(asin, asin,
+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(atan, atan,
+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, 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(ceil, ceil,
+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.")
-FUNC1(cos, cos,
- "cos(x)\n\nReturn the cosine of x (measured in radians).")
-FUNC1(cosh, cosh,
- "cosh(x)\n\nReturn the hyperbolic cosine of x.")
-
-#ifdef MS_WINDOWS
-# define copysign _copysign
-# define HAVE_COPYSIGN 1
-#endif
-#ifdef HAVE_COPYSIGN
FUNC2(copysign, copysign,
- "copysign(x,y)\n\nReturn x with the sign of y.");
-#endif
-
-FUNC1(exp, exp,
+ "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,
+FUNC1(fabs, fabs, 0,
"fabs(x)\n\nReturn the absolute value of the float x.")
-FUNC1(floor, floor,
+FUNC1(floor, floor, 0,
"floor(x)\n\nReturn the floor of x as a float.\n"
"This is the largest integral value <= x.")
-FUNC2(fmod, fmod,
- "fmod(x,y)\n\nReturn fmod(x, y), according to platform C."
- " x % y may differ.")
-FUNC2(hypot, hypot,
- "hypot(x,y)\n\nReturn the Euclidean distance, sqrt(x*x + y*y).")
-FUNC2(pow, pow,
- "pow(x,y)\n\nReturn x**y (x to the power of y).")
-FUNC1(sin, sin,
+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,
+FUNC1(sinh, sinh, 1,
"sinh(x)\n\nReturn the hyperbolic sine of x.")
-FUNC1(sqrt, sqrt,
+FUNC1(sqrt, sqrt, 0,
"sqrt(x)\n\nReturn the square root of x.")
-FUNC1(tan, tan,
+FUNC1(tan, tan, 0,
"tan(x)\n\nReturn the tangent of x (measured in radians).")
-FUNC1(tanh, tanh,
+FUNC1(tanh, tanh, 0,
"tanh(x)\n\nReturn the hyperbolic tangent of x.")
static PyObject *
@@ -172,13 +289,17 @@
double x = PyFloat_AsDouble(arg);
if (x == -1.0 && PyErr_Occurred())
return NULL;
- errno = 0;
- x = frexp(x, &i);
- Py_SET_ERRNO_ON_MATH_ERROR(x);
- if (errno && is_error(x))
- return NULL;
- else
- return Py_BuildValue("(di)", x, i);
+ /* deal with special cases directly, to sidestep platform
+ differences */
+ if (Py_IS_NAN(x) || Py_IS_INFINITY(x) || !x) {
+ i = 0;
+ }
+ else {
+ PyFPE_START_PROTECT("in math_frexp", return 0);
+ x = frexp(x, &i);
+ PyFPE_END_PROTECT(x);
+ }
+ return Py_BuildValue("(di)", x, i);
}
PyDoc_STRVAR(math_frexp_doc,
@@ -191,19 +312,24 @@
static PyObject *
math_ldexp(PyObject *self, PyObject *args)
{
- double x;
+ double x, r;
int exp;
if (! PyArg_ParseTuple(args, "di:ldexp", &x, &exp))
return NULL;
errno = 0;
- PyFPE_START_PROTECT("ldexp", return 0)
- x = ldexp(x, exp);
- PyFPE_END_PROTECT(x)
- Py_SET_ERRNO_ON_MATH_ERROR(x);
- if (errno && is_error(x))
+ PyFPE_START_PROTECT("in math_ldexp", return 0)
+ r = ldexp(x, exp);
+ PyFPE_END_PROTECT(r)
+ if (Py_IS_FINITE(x) && Py_IS_INFINITY(r))
+ errno = ERANGE;
+ /* Windows MSVC8 sets errno = EDOM on ldexp(NaN, i);
+ we unset it to avoid raising a ValueError here. */
+ if (errno == EDOM)
+ errno = 0;
+ if (errno && is_error(r))
return NULL;
else
- return PyFloat_FromDouble(x);
+ return PyFloat_FromDouble(r);
}
PyDoc_STRVAR(math_ldexp_doc,
@@ -216,12 +342,10 @@
if (x == -1.0 && PyErr_Occurred())
return NULL;
errno = 0;
+ PyFPE_START_PROTECT("in math_modf", return 0);
x = modf(x, &y);
- Py_SET_ERRNO_ON_MATH_ERROR(x);
- if (errno && is_error(x))
- return NULL;
- else
- return Py_BuildValue("(dd)", x, y);
+ PyFPE_END_PROTECT(x);
+ return Py_BuildValue("(dd)", x, y);
}
PyDoc_STRVAR(math_modf_doc,
@@ -260,7 +384,7 @@
}
/* Else let libm handle it by itself. */
- return math_1(arg, func);
+ return math_1(arg, func, 0);
}
static PyObject *
@@ -303,6 +427,141 @@
PyDoc_STRVAR(math_log10_doc,
"log10(x) -> the base 10 logarithm of x.");
+static PyObject *
+math_fmod(PyObject *self, PyObject *args)
+{
+ PyObject *ox, *oy;
+ double r, x, y;
+ if (! PyArg_UnpackTuple(args, "fmod", 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;
+ /* fmod(x, +/-Inf) returns x for finite x. */
+ if (Py_IS_INFINITY(y) && Py_IS_FINITE(x))
+ return PyFloat_FromDouble(x);
+ errno = 0;
+ PyFPE_START_PROTECT("in math_fmod", return 0);
+ r = fmod(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;
+ }
+ if (errno && is_error(r))
+ return NULL;
+ else
+ return PyFloat_FromDouble(r);
+}
+
+PyDoc_STRVAR(math_fmod_doc,
+"fmod(x,y)\n\nReturn fmod(x, y), according to platform C."
+" x % y may differ.");
+
+static PyObject *
+math_hypot(PyObject *self, PyObject *args)
+{
+ PyObject *ox, *oy;
+ double r, x, y;
+ if (! PyArg_UnpackTuple(args, "hypot", 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;
+ /* hypot(x, +/-Inf) returns Inf, even if x is a NaN. */
+ if (Py_IS_INFINITY(x))
+ return PyFloat_FromDouble(fabs(x));
+ if (Py_IS_INFINITY(y))
+ return PyFloat_FromDouble(fabs(y));
+ errno = 0;
+ PyFPE_START_PROTECT("in math_hypot", return 0);
+ r = hypot(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);
+}
+
+PyDoc_STRVAR(math_hypot_doc,
+"hypot(x,y)\n\nReturn the Euclidean distance, sqrt(x*x + y*y).");
+
+/* pow can't use math_2, but needs its own wrapper: the problem is
+ that an infinite result can arise either as a result of overflow
+ (in which case OverflowError should be raised) or as a result of
+ e.g. 0.**-5. (for which ValueError needs to be raised.)
+*/
+
+static PyObject *
+math_pow(PyObject *self, PyObject *args)
+{
+ PyObject *ox, *oy;
+ double r, x, y;
+
+ if (! PyArg_UnpackTuple(args, "pow", 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;
+ /* 1**x and x**0 return 1., even if x is a NaN or infinity. */
+ if (x == 1.0 || y == 0.0)
+ return PyFloat_FromDouble(1.);
+ errno = 0;
+ PyFPE_START_PROTECT("in math_pow", return 0);
+ r = pow(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;
+ }
+ /* an infinite result arises either from:
+
+ (A) (+/-0.)**negative,
+ (B) overflow of x**y with both x and y finite (and x nonzero)
+ (C) (+/-inf)**positive, or
+ (D) x**inf with |x| > 1, or x**-inf with |x| < 1.
+
+ In case (A) we want ValueError to be raised. In case (B)
+ OverflowError should be raised. In cases (C) and (D) the infinite
+ result should be returned.
+ */
+ else if (Py_IS_INFINITY(r)) {
+ if (x == 0.)
+ errno = EDOM;
+ else 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);
+}
+
+PyDoc_STRVAR(math_pow_doc,
+"pow(x,y)\n\nReturn x**y (x to the power of y).");
+
static const double degToRad = Py_MATH_PI / 180.0;
static const double radToDeg = 180.0 / Py_MATH_PI;
@@ -356,16 +615,16 @@
"isinf(x) -> bool\n\
Checks if float x is infinite (positive or negative)");
-
static PyMethodDef math_methods[] = {
{"acos", math_acos, METH_O, math_acos_doc},
+ {"acosh", math_acosh, METH_O, math_acosh_doc},
{"asin", math_asin, METH_O, math_asin_doc},
+ {"asinh", math_asinh, METH_O, math_asinh_doc},
{"atan", math_atan, METH_O, math_atan_doc},
{"atan2", math_atan2, METH_VARARGS, math_atan2_doc},
+ {"atanh", math_atanh, METH_O, math_atanh_doc},
{"ceil", math_ceil, METH_O, math_ceil_doc},
-#ifdef HAVE_COPYSIGN
{"copysign", math_copysign, METH_VARARGS, math_copysign_doc},
-#endif
{"cos", math_cos, METH_O, math_cos_doc},
{"cosh", math_cosh, METH_O, math_cosh_doc},
{"degrees", math_degrees, METH_O, math_degrees_doc},
@@ -379,6 +638,7 @@
{"isnan", math_isnan, METH_O, math_isnan_doc},
{"ldexp", math_ldexp, METH_VARARGS, math_ldexp_doc},
{"log", math_log, METH_VARARGS, math_log_doc},
+ {"log1p", math_log1p, METH_O, math_log1p_doc},
{"log10", math_log10, METH_O, math_log10_doc},
{"modf", math_modf, METH_O, math_modf_doc},
{"pow", math_pow, METH_VARARGS, math_pow_doc},
@@ -400,27 +660,15 @@
PyMODINIT_FUNC
initmath(void)
{
- PyObject *m, *d, *v;
+ PyObject *m;
m = Py_InitModule3("math", math_methods, module_doc);
if (m == NULL)
goto finally;
- d = PyModule_GetDict(m);
- if (d == NULL)
- goto finally;
- if (!(v = PyFloat_FromDouble(Py_MATH_PI)))
- goto finally;
- if (PyDict_SetItemString(d, "pi", v) < 0)
- goto finally;
- Py_DECREF(v);
+ PyModule_AddObject(m, "pi", PyFloat_FromDouble(Py_MATH_PI));
+ PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E));
- if (!(v = PyFloat_FromDouble(Py_MATH_E)))
- goto finally;
- if (PyDict_SetItemString(d, "e", v) < 0)
- goto finally;
- Py_DECREF(v);
-
- finally:
+ finally:
return;
}