What it is Numexpr?
===================
Numexpr is a fast numerical expression evaluator for NumPy. With it,
expressions that operate on arrays (like "3*a+4*b") are accelerated
and use less memory than doing the same calculation in Python.
In addition, its multi-threaded capabilities can make use of all your
cores, which may accelerate computations, most specially if they are
not memory-bounded (e.g. those using transcendental functions).
Last but not least, numexpr can make use of Intel's VML (Vector Math
Library, normally integrated in its Math Kernel Library, or MKL).
This allows further acceleration of transcendent expressions.
Examples of use
===============
>>> import numpy as np
>>> import numexpr as ne
>>> a = np.arange(1e6) # Choose large arrays
>>> b = np.arange(1e6)
>>> ne.evaluate("a + 1") # a simple expression
array([ 1.00000000e+00, 2.00000000e+00, 3.00000000e+00, ...,
9.99998000e+05, 9.99999000e+05, 1.00000000e+06])
>>> ne.evaluate('a*b-4.1*a > 2.5*b') # a more complex one
array([False, False, False, ..., True, True, True], dtype=bool)
>>> ne.evaluate("sin(a) + arcsinh(a/b)") # you can also use functions
array([ NaN, 1.72284457, 1.79067101, ..., 1.09567006,
0.17523598, -0.09597844])
>>> s = np.array(['abba', 'abbb', 'abbcdef'])
>>> ne.evaluate("'abba' == s") # string arrays are supported too
array([ True, False, False], dtype=bool)
Datatypes supported internally
==============================
Numexpr operates internally only with the following types:
* 8-bit boolean (bool)
* 32-bit signed integer (int or int32)
* 64-bit signed integer (long or int64)
* 32-bit single-precision floating point number (float or float32)
* 64-bit, double-precision floating point number (double or float64)
* 2x64-bit, double-precision complex number (complex or complex128)
* Raw string of bytes (str)
If the arrays in the expression does not match any of these types,
they will be upcasted to one of the above types (following the usual
type inference rules, see below). Have this in mind when doing
estimations about the memory consumption during the computation of
your expressions.
Also, the types in Numexpr conditions are somewhat stricter than those
of Python. For instance, the only valid constants for booleans are
`True` and `False`, and they are never automatically cast to integers.
Casting rules
=============
Casting rules in Numexpr follow closely those of NumPy. However, for
implementation reasons, there are some known exceptions to this rule,
namely:
* When an array with type `int8`, `uint8`, `int16` or `uint16` is
used inside Numexpr, it is internally upcasted to an `int` (or
`int32` in NumPy notation).
* When an array with type `uint32` is used inside Numexpr, it is
internally upcasted to a `long` (or `int64` in NumPy notation).
* A floating point function (e.g. `sin`) acting on `int8` or
`int16` types returns a `float64` type, instead of the `float32`
that is returned by NumPy functions. This is mainly due to the
absence of native `int8` or `int16` types in Numexpr.
* In operations implying a scalar and an array, the normal rules
of casting are used in Numexpr, in contrast with NumPy, where
array types takes priority. For example, if 'a' is an array of
type `float32` and 'b' is an scalar of type `float64` (or Python
`float` type, which is equivalent), then 'a*b' returns a
`float64` in Numexpr, but a `float32` in NumPy (i.e. array
operands take priority in determining the result type). If you
need to keep the result a `float32`, be sure you use a `float32`
scalar too.
Supported operators
===================
Numexpr supports the set of operators listed below:
* Logical operators: &, |, ~
* Comparison operators: <, <=, ==, !=, >=, >
* Unary arithmetic operators: -
* Binary arithmetic operators: +, -, *, /, **, %
Supported functions
===================
The next are the current supported set:
* where(bool, number1, number2): number
Number1 if the bool condition is true, number2 otherwise.
* {sin,cos,tan}(float|complex): float|complex
Trigonometric sine, cosine or tangent.
* {arcsin,arccos,arctan}(float|complex): float|complex
Trigonometric inverse sine, cosine or tangent.
* arctan2(float1, float2): float
Trigonometric inverse tangent of float1/float2.
* {sinh,cosh,tanh}(float|complex): float|complex
Hyperbolic sine, cosine or tangent.
* {arcsinh,arccosh,arctanh}(float|complex): float|complex
Hyperbolic inverse sine, cosine or tangent.
* {log,log10,log1p}(float|complex): float|complex
Natural, base-10 and log(1+x) logarithms.
* {exp,expm1}(float|complex): float|complex
Exponential and exponential minus one.
* sqrt(float|complex): float|complex
Square root.
* abs(float|complex): float|complex
Absolute value.
* {real,imag}(complex): float
Real or imaginary part of complex.
* complex(float, float): complex
Complex from real and imaginary parts.
.. Notes:
+ `abs()` for complex inputs returns a ``complex`` output too. This
is a departure from NumPy where a ``float`` is returned instead.
However, Numexpr is not flexible enough yet so as to allow this to
happen. Meanwhile, if you want to mimic NumPy behaviour, you may
want to select the real part via the ``real`` function
(e.g. "real(abs(cplx))") or via the ``real`` selector
(e.g. "abs(cplx).real").
More functions can be added if you need them.
Supported reduction operations
==============================
The next are the current supported set:
* sum(number, axis=None): Sum of array elements over a given axis.
Negative axis are not supported.
* prod(number, axis=None): Product of array elements over a given
axis. Negative axis are not supported.
General routines
================
* evaluate(expression, local_dict=None, global_dict=None,
out=None, order='K', casting='safe', **kwargs):
Evaluate a simple array expression element-wise. See docstrings
for more info on parameters. Also, see examples above.
* test(): Run all the tests in the test suite.
* print_versions(): Print the versions of software that numexpr
relies on.
* set_num_threads(nthreads): Sets a number of threads to be used in
operations. Returns the previous setting for the number of
threads. During initialization time Numexpr sets this number to
the number of detected cores in the system (see
`detect_number_of_cores()`).
If you are using Intel's VML, you may want to use
`set_vml_num_threads(nthreads)` to perform the parallel job with
VML instead. However, you should get very similar performance
with VML-optimized functions, and VML's parallelizer cannot deal
with common expresions like `(x+1)*(x-2)`, while Numexpr's one
can.
* detect_number_of_cores(): Detects the number of cores in the
system.
Intel's VML specific support routines
=====================================
When compiled with Intel's VML (Vector Math Library), you will be able
to use some additional functions for controlling its use. These are:
* set_vml_accuracy_mode(mode): Set the accuracy for VML operations.
The `mode` parameter can take the values:
- 'low': Equivalent to VML_LA - low accuracy VML functions are called
- 'high': Equivalent to VML_HA - high accuracy VML functions are called
- 'fast': Equivalent to VML_EP - enhanced performance VML functions are called
It returns the previous mode.
This call is equivalent to the `vmlSetMode()` in the VML library.
See:
http://www.intel.com/software/products/mkl/docs/webhelp/vml/vml_DataTypesAccuracyModes.html
for more info on the accuracy modes.
* set_vml_num_threads(nthreads): Suggests a maximum number of
threads to be used in VML operations.
This function is equivalent to the call
`mkl_domain_set_num_threads(nthreads, MKL_VML)` in the MKL library.
See:
http://www.intel.com/software/products/mkl/docs/webhelp/support/functn_mkl_domain_set_num_threads.html
for more info about it.
* get_vml_version(): Get the VML/MKL library version.
How Numexpr can achieve such a high performance?
================================================
The main reason why Numexpr achieves better performance than NumPy (or
even than plain C code) is that it avoids the creation of whole
temporaries for keeping intermediate results, so saving memory
bandwidth (the main bottleneck in many computations in nowadays
computers). Due to this, it works best with arrays that are large
enough (typically larger than processor caches).
Briefly, it works as follows. Numexpr parses the expression into its
own op-codes, that will be used by the integrated computing virtual
machine. Then, the array operands are split in small chunks (that
easily fit in the cache of the CPU) and passed to the virtual
machine. Then, the computational phase starts, and the virtual machine
applies the op-code operations for each chunk, saving the outcome in
the resulting array. It is worth noting that all the temporaries and
constants in the expression are kept in the same small chunk sizes
than the operand ones, avoiding additional memory (and most specially,
memory bandwidth) waste.
The result is that Numexpr can get the most of your machine computing
capabilities for array-wise computations. Just to give you an idea of
its performance, common speed-ups with regard to NumPy are usually
between 0.95x (for very simple expressions, like ’a + 1’) and 4x (for
relatively complex ones, like 'a*b-4.1*a > 2.5*b'), although much
higher speed-ups can be achieved (up to 15x can be seen in not too
esoteric expressions) because this depends on the kind of the
operations and how many operands participates in the expression. Of
course, Numexpr will perform better (in comparison with NumPy) with
larger matrices, i.e. typically those that does not fit in the cache
of your CPU. In order to get a better idea on the different speed-ups
that can be achieved for your own platform, you may want to run the
benchmarks in the directory bench/.
See more info about how Numexpr works in:
http://code.google.com/p/numexpr/wiki/Overview
Authors
=======
See AUTHORS.txt
License
=======
Numexpr is distributed under the MIT license (see LICENSE.txt file).
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