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