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| """ | |
| Machine arithmetics - determine the parameters of the | |
| floating-point arithmetic system | |
| Author: Pearu Peterson, September 2003 | |
| """ | |
| __all__ = ['MachAr'] | |
| from numpy.core.fromnumeric import any | |
| from numpy.core._ufunc_config import errstate | |
| from numpy.core.overrides import set_module | |
| # Need to speed this up...especially for longfloat | |
| class MachAr: | |
| """ | |
| 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] positive floating point number with full precision). | |
| 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()) | |