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"""
Machine arithmetics - determine the parameters of the
floating-point arithmetic system
Author: Pearu Peterson, September 2003
"""
from __future__ import division, absolute_import, print_function
__all__ = ['MachAr']
from numpy.core.fromnumeric import any
from numpy.core.numeric import errstate
# Need to speed this up...especially for longfloat
class MachAr(object):
"""
Diagnosing machine parameters.
Attributes
----------
ibeta : int
Radix in which numbers are represented.
it : int
Number of base-`ibeta` digits in the floating point mantissa M.
machep : int
Exponent of the smallest (most negative) power of `ibeta` that,
added to 1.0, gives something different from 1.0
eps : float
Floating-point number ``beta**machep`` (floating point precision)
negep : int
Exponent of the smallest power of `ibeta` that, subtracted
from 1.0, gives something different from 1.0.
epsneg : float
Floating-point number ``beta**negep``.
iexp : int
Number of bits in the exponent (including its sign and bias).
minexp : int
Smallest (most negative) power of `ibeta` consistent with there
being no leading zeros in the mantissa.
xmin : float
Floating point number ``beta**minexp`` (the smallest [in
magnitude] usable floating value).
maxexp : int
Smallest (positive) power of `ibeta` that causes overflow.
xmax : float
``(1-epsneg) * beta**maxexp`` (the largest [in magnitude]
usable floating value).
irnd : int
In ``range(6)``, information on what kind of rounding is done
in addition, and on how underflow is handled.
ngrd : int
Number of 'guard digits' used when truncating the product
of two mantissas to fit the representation.
epsilon : float
Same as `eps`.
tiny : float
Same as `xmin`.
huge : float
Same as `xmax`.
precision : float
``- int(-log10(eps))``
resolution : float
``- 10**(-precision)``
Parameters
----------
float_conv : function, optional
Function that converts an integer or integer array to a float
or float array. Default is `float`.
int_conv : function, optional
Function that converts a float or float array to an integer or
integer array. Default is `int`.
float_to_float : function, optional
Function that converts a float array to float. Default is `float`.
Note that this does not seem to do anything useful in the current
implementation.
float_to_str : function, optional
Function that converts a single float to a string. Default is
``lambda v:'%24.16e' %v``.
title : str, optional
Title that is printed in the string representation of `MachAr`.
See Also
--------
finfo : Machine limits for floating point types.
iinfo : Machine limits for integer types.
References
----------
.. [1] Press, Teukolsky, Vetterling and Flannery,
"Numerical Recipes in C++," 2nd ed,
Cambridge University Press, 2002, p. 31.
"""
def __init__(self, float_conv=float,int_conv=int,
float_to_float=float,
float_to_str=lambda v:'%24.16e' % v,
title='Python floating point number'):
"""
float_conv - convert integer to float (array)
int_conv - convert float (array) to integer
float_to_float - convert float array to float
float_to_str - convert array float to str
title - description of used floating point numbers
"""
# We ignore all errors here because we are purposely triggering
# underflow to detect the properties of the runninng arch.
with errstate(under='ignore'):
self._do_init(float_conv, int_conv, float_to_float, float_to_str, title)
def _do_init(self, float_conv, int_conv, float_to_float, float_to_str, title):
max_iterN = 10000
msg = "Did not converge after %d tries with %s"
one = float_conv(1)
two = one + one
zero = one - one
# Do we really need to do this? Aren't they 2 and 2.0?
# Determine ibeta and beta
a = one
for _ in range(max_iterN):
a = a + a
temp = a + one
temp1 = temp - a
if any(temp1 - one != zero):
break
else:
raise RuntimeError(msg % (_, one.dtype))
b = one
for _ in range(max_iterN):
b = b + b
temp = a + b
itemp = int_conv(temp-a)
if any(itemp != 0):
break
else:
raise RuntimeError(msg % (_, one.dtype))
ibeta = itemp
beta = float_conv(ibeta)
# Determine it and irnd
it = -1
b = one
for _ in range(max_iterN):
it = it + 1
b = b * beta
temp = b + one
temp1 = temp - b
if any(temp1 - one != zero):
break
else:
raise RuntimeError(msg % (_, one.dtype))
betah = beta / two
a = one
for _ in range(max_iterN):
a = a + a
temp = a + one
temp1 = temp - a
if any(temp1 - one != zero):
break
else:
raise RuntimeError(msg % (_, one.dtype))
temp = a + betah
irnd = 0
if any(temp-a != zero):
irnd = 1
tempa = a + beta
temp = tempa + betah
if irnd == 0 and any(temp-tempa != zero):
irnd = 2
# Determine negep and epsneg
negep = it + 3
betain = one / beta
a = one
for i in range(negep):
a = a * betain
b = a
for _ in range(max_iterN):
temp = one - a
if any(temp-one != zero):
break
a = a * beta
negep = negep - 1
# Prevent infinite loop on PPC with gcc 4.0:
if negep < 0:
raise RuntimeError("could not determine machine tolerance "
"for 'negep', locals() -> %s" % (locals()))
else:
raise RuntimeError(msg % (_, one.dtype))
negep = -negep
epsneg = a
# Determine machep and eps
machep = - it - 3
a = b
for _ in range(max_iterN):
temp = one + a
if any(temp-one != zero):
break
a = a * beta
machep = machep + 1
else:
raise RuntimeError(msg % (_, one.dtype))
eps = a
# Determine ngrd
ngrd = 0
temp = one + eps
if irnd == 0 and any(temp*one - one != zero):
ngrd = 1
# Determine iexp
i = 0
k = 1
z = betain
t = one + eps
nxres = 0
for _ in range(max_iterN):
y = z
z = y*y
a = z*one # Check here for underflow
temp = z*t
if any(a+a == zero) or any(abs(z) >= y):
break
temp1 = temp * betain
if any(temp1*beta == z):
break
i = i + 1
k = k + k
else:
raise RuntimeError(msg % (_, one.dtype))
if ibeta != 10:
iexp = i + 1
mx = k + k
else:
iexp = 2
iz = ibeta
while k >= iz:
iz = iz * ibeta
iexp = iexp + 1
mx = iz + iz - 1
# Determine minexp and xmin
for _ in range(max_iterN):
xmin = y
y = y * betain
a = y * one
temp = y * t
if any((a + a) != zero) and any(abs(y) < xmin):
k = k + 1
temp1 = temp * betain
if any(temp1*beta == y) and any(temp != y):
nxres = 3
xmin = y
break
else:
break
else:
raise RuntimeError(msg % (_, one.dtype))
minexp = -k
# Determine maxexp, xmax
if mx <= k + k - 3 and ibeta != 10:
mx = mx + mx
iexp = iexp + 1
maxexp = mx + minexp
irnd = irnd + nxres
if irnd >= 2:
maxexp = maxexp - 2
i = maxexp + minexp
if ibeta == 2 and not i:
maxexp = maxexp - 1
if i > 20:
maxexp = maxexp - 1
if any(a != y):
maxexp = maxexp - 2
xmax = one - epsneg
if any(xmax*one != xmax):
xmax = one - beta*epsneg
xmax = xmax / (xmin*beta*beta*beta)
i = maxexp + minexp + 3
for j in range(i):
if ibeta == 2:
xmax = xmax + xmax
else:
xmax = xmax * beta
self.ibeta = ibeta
self.it = it
self.negep = negep
self.epsneg = float_to_float(epsneg)
self._str_epsneg = float_to_str(epsneg)
self.machep = machep
self.eps = float_to_float(eps)
self._str_eps = float_to_str(eps)
self.ngrd = ngrd
self.iexp = iexp
self.minexp = minexp
self.xmin = float_to_float(xmin)
self._str_xmin = float_to_str(xmin)
self.maxexp = maxexp
self.xmax = float_to_float(xmax)
self._str_xmax = float_to_str(xmax)
self.irnd = irnd
self.title = title
# Commonly used parameters
self.epsilon = self.eps
self.tiny = self.xmin
self.huge = self.xmax
import math
self.precision = int(-math.log10(float_to_float(self.eps)))
ten = two + two + two + two + two
resolution = ten ** (-self.precision)
self.resolution = float_to_float(resolution)
self._str_resolution = float_to_str(resolution)
def __str__(self):
fmt = (
'Machine parameters for %(title)s\n'
'---------------------------------------------------------------------\n'
'ibeta=%(ibeta)s it=%(it)s iexp=%(iexp)s ngrd=%(ngrd)s irnd=%(irnd)s\n'
'machep=%(machep)s eps=%(_str_eps)s (beta**machep == epsilon)\n'
'negep =%(negep)s epsneg=%(_str_epsneg)s (beta**epsneg)\n'
'minexp=%(minexp)s xmin=%(_str_xmin)s (beta**minexp == tiny)\n'
'maxexp=%(maxexp)s xmax=%(_str_xmax)s ((1-epsneg)*beta**maxexp == huge)\n'
'---------------------------------------------------------------------\n'
)
return fmt % self.__dict__
if __name__ == '__main__':
print(MachAr())