Numpy core libraries
====================

.. sectionauthor:: David Cournapeau

.. versionadded:: 1.3.0

Starting from numpy 1.3.0, we are working on separating the pure C,
"computational" code from the python dependent code. The goal is twofolds:
making the code cleaner, and enabling code reuse by other extensions outside
numpy (scipy, etc...).

Numpy core math library
-----------------------

The numpy core math library ('npymath') is a first step in this direction. This
library contains most math-related C99 functionality, which can be used on
platforms where C99 is not well supported. The core math functions have the
same API as the C99 ones, except for the npy_* prefix.

The available functions are defined in <numpy/npy_math.h> - please refer to this header when
in doubt.

Floating point classification
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. cvar:: NPY_NAN

    This macro is defined to a NaN (Not a Number), and is guaranteed to have
    the signbit unset ('positive' NaN). The corresponding single and extension
    precision macro are available with the suffix F and L.

.. cvar:: NPY_INFINITY

    This macro is defined to a positive inf. The corresponding single and
    extension precision macro are available with the suffix F and L.

.. cvar:: NPY_PZERO

    This macro is defined to positive zero. The corresponding single and
    extension precision macro are available with the suffix F and L.

.. cvar:: NPY_NZERO

    This macro is defined to negative zero (that is with the sign bit set). The
    corresponding single and extension precision macro are available with the
    suffix F and L.

.. cfunction:: int npy_isnan(x)

    This is a macro, and is equivalent to C99 isnan: works for single, double
    and extended precision, and return a non 0 value is x is a NaN.

.. cfunction:: int npy_isfinite(x)

    This is a macro, and is equivalent to C99 isfinite: works for single,
    double and extended precision, and return a non 0 value is x is neither a
    NaN nor an infinity.

.. cfunction:: int npy_isinf(x)

    This is a macro, and is equivalent to C99 isinf: works for single, double
    and extended precision, and return a non 0 value is x is infinite (positive
    and negative).

.. cfunction:: int npy_signbit(x)

    This is a macro, and is equivalent to C99 signbit: works for single, double
    and extended precision, and return a non 0 value is x has the signbit set
    (that is the number is negative).

.. cfunction:: double npy_copysign(double x, double y)

    This is a function equivalent to C99 copysign: return x with the same sign
    as y. Works for any value, including inf and nan. Single and extended
    precisions are available with suffix f and l.

    .. versionadded:: 1.4.0

Useful math constants
~~~~~~~~~~~~~~~~~~~~~

The following math constants are available in npy_math.h. Single and extended
precision are also available by adding the F and L suffixes respectively.

.. cvar:: NPY_E

    Base of natural logarithm (:math:`e`)

.. cvar:: NPY_LOG2E

    Logarithm to base 2 of the Euler constant (:math:`\frac{\ln(e)}{\ln(2)}`)

.. cvar:: NPY_LOG10E

    Logarithm to base 10 of the Euler constant (:math:`\frac{\ln(e)}{\ln(10)}`)

.. cvar:: NPY_LOGE2

    Natural logarithm of 2 (:math:`\ln(2)`)

.. cvar:: NPY_LOGE10

    Natural logarithm of 10 (:math:`\ln(10)`)

.. cvar:: NPY_PI

    Pi (:math:`\pi`)

.. cvar:: NPY_PI_2

    Pi divided by 2 (:math:`\frac{\pi}{2}`)

.. cvar:: NPY_PI_4

    Pi divided by 4 (:math:`\frac{\pi}{4}`)

.. cvar:: NPY_1_PI

    Reciprocal of pi (:math:`\frac{1}{\pi}`)

.. cvar:: NPY_2_PI

    Two times the reciprocal of pi (:math:`\frac{2}{\pi}`)

.. cvar:: NPY_EULER

    The Euler constant
        :math:`\lim_{n\rightarrow\infty}({\sum_{k=1}^n{\frac{1}{k}}-\ln n})`

Low-level floating point manipulation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Those can be useful for precise floating point comparison.

.. cfunction:: double npy_nextafter(double x, double y)

    This is a function equivalent to C99 nextafter: return next representable
    floating point value from x in the direction of y. Single and extended
    precisions are available with suffix f and l.

    .. versionadded:: 1.4.0

.. cfunction:: double npy_spacing(double x)

    This is a function equivalent to Fortran intrinsic. Return distance between
    x and next representable floating point value from x, e.g. spacing(1) ==
    eps. spacing of nan and +/- inf return nan. Single and extended precisions
    are available with suffix f and l.

    .. versionadded:: 1.4.0

Complex functions
~~~~~~~~~~~~~~~~~

.. versionadded:: 1.4.0

C99-like complex functions have been added. Those can be used if you wish to
implement portable C extensions. Since we still support platforms without C99
complex type, you need to restrict to C90-compatible syntax, e.g.:

.. code-block:: c

        /* a = 1 + 2i \*/
        npy_complex a = npy_cpack(1, 2);
        npy_complex b;

        b = npy_log(a);

Linking against the core math library in an extension
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. versionadded:: 1.4.0

To use the core math library in your own extension, you need to add the npymath
compile and link options to your extension in your setup.py:

        >>> from numpy.distutils.misc_util import get_info
        >>> info = get_info('npymath')
        >>> config.add_extension('foo', sources=['foo.c'], extra_info=info)

In other words, the usage of info is exactly the same as when using blas_info
and co.

Half-precision functions
~~~~~~~~~~~~~~~~~~~~~~~~

.. versionadded:: 2.0.0

The header file <numpy/halffloat.h> provides functions to work with
IEEE 754-2008 16-bit floating point values. While this format is
not typically used for numerical computations, it is useful for
storing values which require floating point but do not need much precision.
It can also be used as an educational tool to understand the nature
of floating point round-off error.

Like for other types, NumPy includes a typedef npy_half for the 16 bit
float.  Unlike for most of the other types, you cannot use this as a
normal type in C, since is is a typedef for npy_uint16.  For example,
1.0 looks like 0x3c00 to C, and if you do an equality comparison
between the different signed zeros, you will get -0.0 != 0.0
(0x8000 != 0x0000), which is incorrect.

For these reasons, NumPy provides an API to work with npy_half values
accessible by including <numpy/halffloat.h> and linking to 'npymath'.
For functions that are not provided directly, such as the arithmetic
operations, the preferred method is to convert to float
or double and back again, as in the following example.

.. code-block:: c

        npy_half sum(int n, npy_half *array) {
            float ret = 0;
            while(n--) {
                ret += npy_half_to_float(*array++);
            }
            return npy_float_to_half(ret);
        }

External Links:

* `754-2008 IEEE Standard for Floating-Point Arithmetic`__
* `Half-precision Float Wikipedia Article`__.
* `OpenGL Half Float Pixel Support`__
* `The OpenEXR image format`__.

__ http://ieeexplore.ieee.org/servlet/opac?punumber=4610933
__ http://en.wikipedia.org/wiki/Half_precision_floating-point_format
__ http://www.opengl.org/registry/specs/ARB/half_float_pixel.txt
__ http://www.openexr.com/about.html

.. cvar:: NPY_HALF_ZERO

    This macro is defined to positive zero.

.. cvar:: NPY_HALF_PZERO

    This macro is defined to positive zero.

.. cvar:: NPY_HALF_NZERO

    This macro is defined to negative zero.

.. cvar:: NPY_HALF_ONE

    This macro is defined to 1.0.

.. cvar:: NPY_HALF_NEGONE

    This macro is defined to -1.0.

.. cvar:: NPY_HALF_PINF

    This macro is defined to +inf.

.. cvar:: NPY_HALF_NINF

    This macro is defined to -inf.

.. cvar:: NPY_HALF_NAN

    This macro is defined to a NaN value, guaranteed to have its sign bit unset.

.. cfunction:: float npy_half_to_float(npy_half h)

   Converts a half-precision float to a single-precision float.

.. cfunction:: double npy_half_to_double(npy_half h)

   Converts a half-precision float to a double-precision float.

.. cfunction:: npy_half npy_float_to_half(float f)

   Converts a single-precision float to a half-precision float.  The
   value is rounded to the nearest representable half, with ties going
   to the nearest even.  If the value is too small or too big, the
   system's floating point underflow or overflow bit will be set.

.. cfunction:: npy_half npy_double_to_half(double d)

   Converts a double-precision float to a half-precision float.  The
   value is rounded to the nearest representable half, with ties going
   to the nearest even.  If the value is too small or too big, the
   system's floating point underflow or overflow bit will be set.

.. cfunction:: int npy_half_eq(npy_half h1, npy_half h2)

   Compares two half-precision floats (h1 == h2).

.. cfunction:: int npy_half_ne(npy_half h1, npy_half h2)

   Compares two half-precision floats (h1 != h2).

.. cfunction:: int npy_half_le(npy_half h1, npy_half h2)

   Compares two half-precision floats (h1 <= h2).

.. cfunction:: int npy_half_lt(npy_half h1, npy_half h2)

   Compares two half-precision floats (h1 < h2).

.. cfunction:: int npy_half_ge(npy_half h1, npy_half h2)

   Compares two half-precision floats (h1 >= h2).

.. cfunction:: int npy_half_gt(npy_half h1, npy_half h2)

   Compares two half-precision floats (h1 > h2).

.. cfunction:: int npy_half_eq_nonan(npy_half h1, npy_half h2)

   Compares two half-precision floats that are known to not be NaN (h1 == h2).  If
   a value is NaN, the result is undefined.

.. cfunction:: int npy_half_lt_nonan(npy_half h1, npy_half h2)

   Compares two half-precision floats that are known to not be NaN (h1 < h2).  If
   a value is NaN, the result is undefined.

.. cfunction:: int npy_half_le_nonan(npy_half h1, npy_half h2)

   Compares two half-precision floats that are known to not be NaN (h1 <= h2).  If
   a value is NaN, the result is undefined.

.. cfunction:: int npy_half_iszero(npy_half h)

   Tests whether the half-precision float has a value equal to zero.  This may be slightly
   faster than calling npy_half_eq(h, NPY_ZERO).

.. cfunction:: int npy_half_isnan(npy_half h)

   Tests whether the half-precision float is a NaN.

.. cfunction:: int npy_half_isinf(npy_half h)

   Tests whether the half-precision float is plus or minus Inf.

.. cfunction:: int npy_half_isfinite(npy_half h)

   Tests whether the half-precision float is finite (not NaN or Inf).

.. cfunction:: int npy_half_signbit(npy_half h)

   Returns 1 is h is negative, 0 otherwise.

.. cfunction:: npy_half npy_half_copysign(npy_half x, npy_half y)

    Returns the value of x with the sign bit copied from y.  Works for any value,
    including Inf and NaN.
    
.. cfunction:: npy_half npy_half_spacing(npy_half h)

    This is the same for half-precision float as npy_spacing and npy_spacingf
    described in the low-level floating point section.

.. cfunction:: npy_half npy_half_nextafter(npy_half x, npy_half y)

    This is the same for half-precision float as npy_nextafter and npy_nextafterf
    described in the low-level floating point section.

.. cfunction:: npy_uint16 npy_floatbits_to_halfbits(npy_uint32 f)

   Low-level function which converts a 32-bit single-precision float, stored
   as a uint32, into a 16-bit half-precision float.

.. cfunction:: npy_uint16 npy_doublebits_to_halfbits(npy_uint64 d)

   Low-level function which converts a 64-bit double-precision float, stored
   as a uint64, into a 16-bit half-precision float.

.. cfunction:: npy_uint32 npy_halfbits_to_floatbits(npy_uint16 h)

   Low-level function which converts a 16-bit half-precision float
   into a 32-bit single-precision float, stored as a uint32.

.. cfunction:: npy_uint64 npy_halfbits_to_doublebits(npy_uint16 h)

   Low-level function which converts a 16-bit half-precision float
   into a 64-bit double-precision float, stored as a uint64. 

