Accelerated UFuncs

Variants of some Numpy UFuncs that use Intel’s Vector Math Library (VML) are found in the accelerate.mkl package in the ufuncs object. In comparison to Numpy’s built-in UFuncs, Accelerated UFuncs have the following properties:

Performance
Because Accelerated UFuncs call functions from VML, which is a library optimised for high performance using multiple threads and SIMD instructions, performance may be increased when calling Accelerated UFuncs in comparison to calling Numpy UFuncs. The performance increase will only be obtained for contiguous arguments. For non-contiguous arguments, performance comparable to Numpy’s will be observed.
Accuracy
  • Accelerated UFuncs produce similar results to their Numpy equivalents for the range of finite values, up to a given relative tolerance.
  • The tolerance varies between functions and data types, and is specified for each combination in the following section.
  • For the range of infinite and NaN values, the majority of results computed by Accelerated UFuncs will be equal to those computed by their Numpy counterparts. However, this is not guaranteed; for example, an input for which the Numpy UFunc produces a result of x + inf*j, the equivalent Accelerated UFunc may produce a result of y + inf*j, where x != y.
  • Towards the edge of the domain of a data type (e.g. near 3.4e+38 for float and 1.79e+308 for double and values of similar magnitude at the negative end of the domain) some Accelerated UFuncs may produce results which differ from Numpy UFuncs, or may raise FloatingPointError or ZeroDivisionError exceptions. These functions are marked domain edge warning in the Accuracy column of the tables in the following sections.
  • Some numpy implementations of functions that operate in the complex domain have branch cuts which differ from those in MKL, where this has been identified in testing these functions are marked branch cut warning in the Accuracy column of the tables in the following sections. This warning may also be present in the case of implementations which contain incorrect branch cuts.
  • Denormal input values may be treated as zero by Accelerated UFuncs, and denormal output values may be flushed to zero.
  • The accuracy reported is approximate and related to the least accurate value encountered when testing with contiguous input arguments. Non-contiguous arguments are more directly exposed to system math library implementations and so their accuracy is not commented on.
Exception handling
For the range of finite values, exceptions will not be raised by Accelerated UFuncs, just as they would not by Numpy. For the infinite and NaN ranges, Accelerated UFuncs may raise FloatingPointError or ZeroDivisionError exceptions in cases when Numpy would not, and vice-versa.

Supported functions are described in the following sections.

Arithmetic Functions

Function Type Accuracy
add(x, y) f4 rtol=1.0e-7
f8 rtol=1.0e-15
c8 rtol=1.0e-7
c16 rtol=1.0e-15
subtract(x, y) f4 rtol=1.e-7
f8 rtol=1.e-15
c8 rtol=1.e-7
c16 rtol=1.e-15
square(x) f4 rtol=1.e-7
f8 rtol=1.e-15
multiply(x, y) f4 rtol=1.e-7
f8 rtol=1.e-15
c8 rtol=1.e-6
c16 rtol=1.e-15
absolute(x) f4 rtol=1.e-6
f8 rtol=1.e-15

Power and Root Functions

Function Type Accuracy
reciprocal(x) f4 rtol=1.e-7
f8 rtol=1.e-15
true_divide(x, y) f4 rtol=1.e-7
f8 rtol=1.e-15
c8 rtol=1.e-6
c16 rtol=1.e-15
sqrt(x, y) f4 rtol=1.e-6
f8 rtol=1.e-15
c8 rtol=1.e-6, domain edge warning
c16 rtol=1.e-15, domain edge warning
power(x, y) f4 rtol=1.e-7
f8 rtol=1.e-15
hypot(x, y) f4 rtol=1.e-6
f8 rtol=1.e-15

Exponential and Logarithmic Functions

Function Type Accuracy
exp(x) f4 rtol=1.e-6
f8 rtol=1.e-15
c8 rtol=1.e-6, domain edge warning
c16 rtol=1.e-15, domain edge warning
expm1(x) f4 rtol=1.e-6
f8 rtol=1.e-15
log(x) f4 rtol=1.e-6
f8 rtol=1.e-15
c8 rtol=1.e-5
c16 rtol=1.e-13
log10(x) f4 rtol=1.e-6
f8 rtol=1.e-15
c8 rtol=1.e-5, domain edge warning
c16 rtol=1.e-13, domain edge warning
log1p(x) f4 rtol=1.e-6
f8 rtol=1.e-15

Trigonometric Functions

Function Type Accuracy
cos(x) f4 rtol=1.e-6, domain edge warning
f8 rtol=1.e-15, domain edge warning
c8 rtol=1.e-6, domain edge warning
c16 rtol=1.e-15, domain edge warning
sin(x) f4 rtol=1.e-5, domain edge warning
f8 rtol=1.e-15, domain edge warning
c8 rtol=1.e-6, domain edge warning
c16 rtol=1.e-15, domain edge warning
tan(x) f4 rtol=1.e-6, domain edge warning
f8 rtol=1.e-15, domain edge warning
c8 rtol=5.e-4, domain edge warning
c16 rtol=5.e-14, domain edge warning
arccos(x) f4 rtol=1.e-6, branch cut warning
f8 rtol=1.e-15, branch cut warning
c8
rtol=1.e-5, domain edge warning,
branch cut warning
c16
rtol=1.e-14, domain edge warning,
branch cut warning
arcsin(x) f4 rtol=1.e-6, branch cut warning
f8 rtol=1.e-15, branch cut warning
c8
rtol=1.e-3, domain edge warning,
branch cut warning
c16
rtol=1.e-12, domain edge warning,
branch cut warning
arctan(x) f4 rtol=1.e-6
f8 rtol=1.e-15
c8 rtol=1.e-6
c16 rtol=1.e-15
arctan2(x) f4 rtol=1.e-6
f8 rtol=1.e-15

Hyperbolic Functions

Function Type Accuracy
cosh(x) f4 rtol=1.e-6, domain edge warning
f8 rtol=1.e-15, domain edge warning
c8 rtol=1.e-6, domain edge warning
c16 rtol=1.e-15, domain edge warning
sinh(x) f4 rtol=1.e-6, domain edge warning
f8 rtol=1.e-15, domain edge warning
c8 rtol=1.e-6, domain edge warning
c16 rtol=1.e-15, domain edge warning
tanh(x) f4 rtol=1.e-6, domain edge warning
f8 rtol=1.e-15, domain edge warning
c8 rtol=1.e-6, domain edge warning
c16 rtol=1.e-15, domain edge warning
arccosh(x) f4 rtol=1.e-6
f8 rtol=1.e-15
c8 rtol=1.e-5, domain edge warning
c16 rtol=1.e-14, domain edge warning
arcsinh(x) f4 rtol=1.e-6
f8 rtol=1.e-15
c8 rtol=5.e-5, domain edge warning
c16 rtol=1.e-13, domain edge warning
arctanh(x) f4 rtol=1.e-5, branch cut warning
f8 rtol=5.e-11, branch cut warning
c8
rtol=1.e-4, domain edge warning,
branch cut warning
c16
rtol=5.e-14, domain edge warning,
branch cut warning