Return ``e**x``.
+.. function:: expm1(x)
+
+ Return ``e**x - 1``. For small floats *x*, the subtraction in
+ ``exp(x) - 1`` can result in a significant loss of precision; the
+ :func:`expm1` function provides a way to compute this quantity to
+ full precision::
+
+ >>> from math import exp, expm1
+ >>> exp(1e-5) - 1 # gives result accurate to 11 places
+ 1.0000050000069649e-05
+ >>> expm1(1e-5) # result accurate to full precision
+ 1.0000050000166668e-05
+
+
.. function:: log(x[, base])
With one argument, return the natural logarithm of *x* (to base *e*).
-- thanks to loss of accuracy in 1-x
gam0140 gamma -63.349078729022985 -> 4.1777971677761880e-88
gam0141 gamma -127.45117632943295 -> 1.1831110896236810e-214
+
+-----------------------------------------------------------
+-- expm1: exp(x) - 1, without precision loss for small x --
+-----------------------------------------------------------
+
+-- special values
+expm10000 expm1 0.0 -> 0.0
+expm10001 expm1 -0.0 -> -0.0
+expm10002 expm1 inf -> inf
+expm10003 expm1 -inf -> -1.0
+expm10004 expm1 nan -> nan
+
+-- expm1(x) ~ x for tiny x
+expm10010 expm1 5e-324 -> 5e-324
+expm10011 expm1 1e-320 -> 1e-320
+expm10012 expm1 1e-300 -> 1e-300
+expm10013 expm1 1e-150 -> 1e-150
+expm10014 expm1 1e-20 -> 1e-20
+
+expm10020 expm1 -5e-324 -> -5e-324
+expm10021 expm1 -1e-320 -> -1e-320
+expm10022 expm1 -1e-300 -> -1e-300
+expm10023 expm1 -1e-150 -> -1e-150
+expm10024 expm1 -1e-20 -> -1e-20
+
+-- moderate sized values, where direct evaluation runs into trouble
+expm10100 expm1 1e-10 -> 1.0000000000500000e-10
+expm10101 expm1 -9.9999999999999995e-08 -> -9.9999995000000163e-8
+expm10102 expm1 3.0000000000000001e-05 -> 3.0000450004500034e-5
+expm10103 expm1 -0.0070000000000000001 -> -0.0069755570667648951
+expm10104 expm1 -0.071499208740094633 -> -0.069002985744820250
+expm10105 expm1 -0.063296004180116799 -> -0.061334416373633009
+expm10106 expm1 0.02390954035597756 -> 0.024197665143819942
+expm10107 expm1 0.085637352649044901 -> 0.089411184580357767
+expm10108 expm1 0.5966174947411006 -> 0.81596588596501485
+expm10109 expm1 0.30247206212075139 -> 0.35319987035848677
+expm10110 expm1 0.74574727375889516 -> 1.1080161116737459
+expm10111 expm1 0.97767512926555711 -> 1.6582689207372185
+expm10112 expm1 0.8450154566787712 -> 1.3280137976535897
+expm10113 expm1 -0.13979260323125264 -> -0.13046144381396060
+expm10114 expm1 -0.52899322039643271 -> -0.41080213643695923
+expm10115 expm1 -0.74083261478900631 -> -0.52328317124797097
+expm10116 expm1 -0.93847766984546055 -> -0.60877704724085946
+expm10117 expm1 10.0 -> 22025.465794806718
+expm10118 expm1 27.0 -> 532048240600.79865
+expm10119 expm1 123 -> 2.6195173187490626e+53
+expm10120 expm1 -12.0 -> -0.99999385578764666
+expm10121 expm1 -35.100000000000001 -> -0.99999999999999944
+
+-- extreme negative values
+expm10201 expm1 -37.0 -> -0.99999999999999989
+expm10200 expm1 -38.0 -> -1.0
+expm10210 expm1 -710.0 -> -1.0
+-- the formula expm1(x) = 2 * sinh(x/2) * exp(x/2) doesn't work so
+-- well when exp(x/2) is subnormal or underflows to zero; check we're
+-- not using it!
+expm10211 expm1 -1420.0 -> -1.0
+expm10212 expm1 -1450.0 -> -1.0
+expm10213 expm1 -1500.0 -> -1.0
+expm10214 expm1 -1e50 -> -1.0
+expm10215 expm1 -1.79e308 -> -1.0
+
+-- extreme positive values
+expm10300 expm1 300 -> 1.9424263952412558e+130
+expm10301 expm1 700 -> 1.0142320547350045e+304
+expm10302 expm1 709.78271289328393 -> 1.7976931346824240e+308
+expm10303 expm1 709.78271289348402 -> inf overflow
+expm10304 expm1 1000 -> inf overflow
+expm10305 expm1 1e50 -> inf overflow
+expm10306 expm1 1.79e308 -> inf overflow
if math.isnan(expected) and math.isnan(got):
continue
if not math.isnan(expected) and not math.isnan(got):
- # we use different closeness criteria for
- # different functions.
- if fn == 'gamma':
- accuracy_failure = ulps_check(expected, got, 20)
- elif fn == 'lgamma':
+ if fn == 'lgamma':
+ # we use a weaker accuracy test for lgamma;
+ # lgamma only achieves an absolute error of
+ # a few multiples of the machine accuracy, in
+ # general.
accuracy_failure = acc_check(expected, got,
rel_err = 5e-15,
abs_err = 5e-15)
else:
- raise ValueError("don't know how to check accuracy "
- "for this function")
+ accuracy_failure = ulps_check(expected, got, 20)
if accuracy_failure is None:
continue
- Issue #7078: Set struct.__doc__ from _struct.__doc__.
-- Issue #3366: Add gamma, lgamma functions to math module.
+- Issue #3366: Add expm1, gamma, lgamma functions to math module.
- Issue #6877: It is now possible to link the readline extension to the
libedit readline emulation on OSX 10.5 or later.
#array arraymodule.c # array objects
#cmath cmathmodule.c # -lm # complex math library functions
-#math mathmodule.c # -lm # math library functions, e.g. sin()
+#math mathmodule.c _math.c # -lm # math library functions, e.g. sin()
#_struct _struct.c # binary structure packing/unpacking
#time timemodule.c # -lm # time operations and variables
#operator operator.c # operator.add() and similar goodies
--- /dev/null
+/* Definitions of some C99 math library functions, for those platforms
+ that don't implement these functions already. */
+
+#include <float.h>
+#include <math.h>
+
+/* Mathematically, expm1(x) = exp(x) - 1. The expm1 function is designed
+ to avoid the significant loss of precision that arises from direct
+ evaluation of the expression exp(x) - 1, for x near 0. */
+
+double
+_Py_expm1(double x)
+{
+ /* For abs(x) >= log(2), it's safe to evaluate exp(x) - 1 directly; this
+ also works fine for infinities and nans.
+
+ For smaller x, we can use a method due to Kahan that achieves close to
+ full accuracy.
+ */
+
+ if (fabs(x) < 0.7) {
+ double u;
+ u = exp(x);
+ if (u == 1.0)
+ return x;
+ else
+ return (u - 1.0) * x / log(u);
+ }
+ else
+ return exp(x) - 1.0;
+}
--- /dev/null
+double _Py_expm1(double x);
+
+#ifdef HAVE_EXPM1
+#define m_expm1 expm1
+#else
+/* if the system doesn't have expm1, use the substitute
+ function defined in Modules/_math.c. */
+#define m_expm1 _Py_expm1
+#endif
*/
#include "Python.h"
+#include "_math.h"
#include "longintrepr.h" /* just for SHIFT */
#ifdef _OSF_SOURCE
"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(expm1, m_expm1, 1,
+ "expm1(x)\n\nReturn exp(x)-1.\n"
+ "This function avoids the loss of precision involved in the direct "
+ "evaluation of exp(x)-1 for small x.")
FUNC1(fabs, fabs, 0,
"fabs(x)\n\nReturn the absolute value of the float x.")
{"cosh", math_cosh, METH_O, math_cosh_doc},
{"degrees", math_degrees, METH_O, math_degrees_doc},
{"exp", math_exp, METH_O, math_exp_doc},
+ {"expm1", math_expm1, METH_O, math_expm1_doc},
{"fabs", math_fabs, METH_O, math_fabs_doc},
{"factorial", math_factorial, METH_O, math_factorial_doc},
{"floor", math_floor, METH_O, math_floor_doc},
# End Source File\r
# Begin Source File\r
\r
+SOURCE=..\..\Modules\_math.c\r
+# End Source File\r
+# Begin Source File\r
+\r
SOURCE=..\..\Modules\_pickle.c\r
# End Source File\r
# Begin Source File\r
<File
RelativePath="..\..\Modules\_lsprof.c">
</File>
+ <File
+ RelativePath="..\..\Modules\_math.c">
+ </File>
<File
RelativePath="..\..\Modules\_pickle.c">
</File>
RelativePath="..\..\Modules\_lsprof.c"
>
</File>
+ <File
+ RelativePath="..\..\Modules\_math.c"
+ >
+ </File>
+ <File
+ RelativePath="..\..\Modules\_math.h"
+ >
+ </File>
<File
RelativePath="..\..\Modules\_pickle.c"
>
RelativePath="..\Modules\_lsprof.c"
>
</File>
+ <File
+ RelativePath="..\Modules\_math.c"
+ >
+ </File>
+ <File
+ RelativePath="..\Modules\_math.h"
+ >
+ </File>
<File
RelativePath="..\Modules\_pickle.c"
>
libraries=math_libs) )
# math library functions, e.g. sin()
- exts.append( Extension('math', ['mathmodule.c'],
+ exts.append( Extension('math', ['mathmodule.c', '_math.c'],
libraries=math_libs) )
# time operations and variables
exts.append( Extension('time', ['timemodule.c'],
projects / python / commitdiff