Class: Flt::DecNum
- Includes:
- AuxiliarFunctions
- Defined in:
- lib/flt/dec_num.rb,
lib/flt/trigonometry.rb
Overview
DecNum arbitrary precision floating point number. This implementation of DecNum is based on the Decimal module of Python, written by Eric Price, Facundo Batista, Raymond Hettinger, Aahz and Tim Peters.
Defined Under Namespace
Modules: AuxiliarFunctions, CMath, Math, Trigonometry Classes: Context
Constant Summary collapse
- DefaultContext =
the DefaultContext is the base for new contexts; it can be changed.
DecNum::Context.new( :exact=>false, :precision=>28, :rounding=>:half_even, :emin=> -999999999, :emax=>+999999999, :flags=>[], :traps=>[DivisionByZero, Overflow, InvalidOperation], :ignored_flags=>[], :capitals=>true, :clamp=>true)
- BasicContext =
DecNum::Context.new(DefaultContext, :precision=>9, :rounding=>:half_up, :traps=>[DivisionByZero, Overflow, InvalidOperation, Clamped, Underflow], :flags=>[])
- ExtendedContext =
DecNum::Context.new(DefaultContext, :precision=>9, :rounding=>:half_even, :traps=>[], :flags=>[], :clamp=>false)
Constants included from AuxiliarFunctions
AuxiliarFunctions::LOG10_LB_CORRECTION, AuxiliarFunctions::NUMBER_OF_DIGITS_MAX_VALID_LOG
Constants inherited from Num
Num::EXCEPTIONS, Num::ROUND_05UP, Num::ROUND_CEILING, Num::ROUND_DOWN, Num::ROUND_FLOOR, Num::ROUND_HALF_DOWN, Num::ROUND_HALF_EVEN, Num::ROUND_HALF_UP, Num::ROUND_UP
Constants included from Num::AuxiliarFunctions
Num::AuxiliarFunctions::EXP_INC, Num::AuxiliarFunctions::LOG10_LB_CORRECTION, Num::AuxiliarFunctions::LOG10_MULT, Num::AuxiliarFunctions::LOG2_LB_CORRECTION, Num::AuxiliarFunctions::LOG2_MULT, Num::AuxiliarFunctions::LOG_PREC_INC, Num::AuxiliarFunctions::LOG_RADIX_EXTRA, Num::AuxiliarFunctions::LOG_RADIX_INC
Constants included from Support::AuxiliarFunctions
Support::AuxiliarFunctions::NBITS_BLOCK, Support::AuxiliarFunctions::NBITS_LIMIT, Support::AuxiliarFunctions::NDIGITS_BLOCK, Support::AuxiliarFunctions::NDIGITS_LIMIT
Class Method Summary collapse
-
.int_div_radix_power(x, n) ⇒ Object
Divide by an integral power of the base: x/(radix**n) for x,n integer; returns an integer.
-
.int_mult_radix_power(x, n) ⇒ Object
Multiply by an integral power of the base: x*(radix**n) for x,n integer; returns an integer.
-
.int_radix_power(n) ⇒ Object
Integral power of the base: radix**n for integer n; returns an integer.
-
.radix ⇒ Object
Numerical base of DecNum.
Instance Method Summary collapse
-
#_ln_exp_bound ⇒ Object
Compute a lower bound for the adjusted exponent of self.ln().
-
#_log10_exp_bound ⇒ Object
Compute a lower bound for the adjusted exponent of self.log10() In other words, find r such that self.log10() >= 10**r.
-
#_power_exact(other, p) ⇒ Object
Attempt to compute self**other exactly Given Decimals self and other and an integer p, attempt to compute an exact result for the power self**other, with p digits of precision.
-
#_power_modulo(other, modulo, context = nil) ⇒ Object
Power-modulo: self._power_modulo(other, modulo) == (self**other) % modulo This is equivalent to Python’s 3-argument version of pow().
-
#exp(context = nil) ⇒ Object
Exponential function.
-
#initialize(*args) ⇒ DecNum
constructor
A DecNum value can be defined by: * A String containing a text representation of the number * An Integer * A Rational * A Value of a type for which conversion is defined in the context.
-
#ln(context = nil) ⇒ Object
Returns the natural (base e) logarithm.
-
#log10(context = nil) ⇒ Object
Returns the base 10 logarithm.
- #number_of_digits ⇒ Object
-
#power(other, modulo = nil, context = nil) ⇒ Object
Raises to the power of x, to modulo if given.
Methods included from AuxiliarFunctions
_dexp, _div_nearest, _dlog, _dlog10, _dpower, _iexp, _ilog, _log10_digits, _log10_lb, _number_of_digits, _rshift_nearest, _sqrt_nearest, dexp
Methods inherited from Num
#%, #*, #**, #+, #+@, #-, #-@, #/, #<, #<=, #<=>, #==, #>, #>=, Context, Flags, Num, [], #_abs, #_check_nans, #_fix, #_fix_nan, #_neg, #_pos, #_rescale, #_watched_rescale, #abs, #add, #adjusted_exponent, base_coercible_types, base_conversions, ccontext, #ceil, #coefficient, #coerce, #compare, context, context=, #convert_to, #copy_abs, #copy_negate, #copy_sign, define_context, #digits, #div, #divide, #divide_int, #divmod, #divrem, #eql?, #even?, #exponent, #finite?, #floor, #fma, #fraction_part, #fractional_exponent, #hash, #infinite?, infinity, #inspect, #integer_part, #integral?, #integral_exponent, #integral_significand, local_context, #log, #log2, #logb, math, #minus, #modulo, #multiply, nan, #nan?, #next_minus, #next_plus, #next_toward, #nonzero?, #normal?, #normalize, num_class, #num_class, #number_class, #odd?, one_half, #plus, #qnan?, #quantize, #rationalize, #reduce, #reduced_exponent, #remainder, #remainder_near, #rescale, #round, #same_quantum?, #scaleb, #scientific_exponent, set_context, #sign, #snan?, #special?, #split, #sqrt, #subnormal?, #subtract, #to_f, #to_i, #to_int_scale, #to_integral_exact, #to_integral_value, #to_r, #to_s, #truncate, #ulp, zero, #zero?
Methods included from Support
FlagValues, Flags, adjust_digits, simplified_round_mode
Methods included from Num::AuxiliarFunctions
_convert, _div_nearest, _exp, _iexp, _ilog, _log, _log_radix_digits, _log_radix_lb, _log_radix_mult, _normalize, _number_of_digits, _parser, _power, _rshift_nearest, _sqrt_nearest, log10_lb, log2_lb
Methods included from Support::AuxiliarFunctions
_nbits, _ndigits, detect_float_rounding
Methods inherited from Numeric
Constructor Details
#initialize(*args) ⇒ DecNum
A DecNum value can be defined by:
-
A String containing a text representation of the number
-
An Integer
-
A Rational
-
A Value of a type for which conversion is defined in the context.
-
Another DecNum.
-
A sign, coefficient and exponent (either as separate arguments, as an array or as a Hash with symbolic keys), or a signed coefficient and an exponent. This is the internal representation of Num, as returned by Num#split. The sign is +1 for plus and -1 for minus; the coefficient and exponent are integers, except for special values which are defined by :inf, :nan or :snan for the exponent.
An optional Context can be passed after the value-definint argument to override the current context and options can be passed in a last hash argument; alternatively context options can be overriden by options of the hash argument.
When the number is defined by a numeric literal (a String), it can be followed by a symbol that specifies the mode used to convert the literal to a floating-point value:
-
:free is currently the default for all cases. The precision of the input literal (including trailing zeros) is preserved and the precision of the context is ignored. When the literal is in base 10, (which is the case by default), the literal is preserved exactly. Otherwise, all significative digits that can be derived from the literal are generanted, significative meaning here that if the digit is changed and the value converted back to a literal of the same base and precision, the original literal will not be obtained.
-
:short is a variation of :free in which only the minimun number of digits that are necessary to produce the original literal when the value is converted back with the same original precision.
-
:fixed will round and normalize the value to the precision specified by the context (normalize meaning that exaclty the number of digits specified by the precision will be generated, even if the original literal has fewer digits.) This may fail returning NaN (and raising Inexact) if the context precision is :exact, but not if the floating-point radix is a multiple of the input base.
Options that can be passed for construction from literal:
-
:base is the numeric base of the input, 10 by default.
The Flt.DecNum() constructor admits the same parameters and can be used as a shortcut for DecNum creation. Examples:
DecNum('0.1000') # -> 0.1000
DecNum('0.12345') # -> 0.12345
DecNum('1.2345E-1') # -> 0.12345
DecNum('0.1000', :short) # -> 0.1
DecNum('0.1000',:fixed, :precision=>20) # -> 0.10000000000000000000
DecNum('0.12345',:fixed, :precision=>20) # -> 0.12345000000000000000
DecNum('0.100110E3', :base=>2) # -> 4.8
DecNum('0.1E-5', :free, :base=>2) # -> 0.016
DecNum('0.1E-5', :short, :base=>2) # -> 0.02
DecNum('0.1E-5', :fixed, :base=>2, :exact=>true) # -> 0.015625
DecNum('0.1E-5', :fixed, :base=>2) # -> 0.01562500000000000000000000000
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# File 'lib/flt/dec_num.rb', line 115 def initialize(*args) super(*args) end |
Class Method Details
.int_div_radix_power(x, n) ⇒ Object
Divide by an integral power of the base: x/(radix**n) for x,n integer; returns an integer.
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# File 'lib/flt/dec_num.rb', line 30 def int_div_radix_power(x,n) n < 0 ? (x * (10**(-n))) : (x / (10**n)) end |
.int_mult_radix_power(x, n) ⇒ Object
Multiply by an integral power of the base: x*(radix**n) for x,n integer; returns an integer.
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# File 'lib/flt/dec_num.rb', line 24 def int_mult_radix_power(x,n) n < 0 ? (x / (10**(-n))) : (x * (10**n)) end |
.int_radix_power(n) ⇒ Object
Integral power of the base: radix**n for integer n; returns an integer.
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# File 'lib/flt/dec_num.rb', line 18 def int_radix_power(n) 10**n end |
.radix ⇒ Object
Numerical base of DecNum.
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# File 'lib/flt/dec_num.rb', line 13 def radix 10 end |
Instance Method Details
#_ln_exp_bound ⇒ Object
Compute a lower bound for the adjusted exponent of self.ln(). In other words, compute r such that self.ln() >= 10**r. Assumes that self is finite and positive and that self != 1.
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# File 'lib/flt/dec_num.rb', line 789 def _ln_exp_bound # for 0.1 <= x <= 10 we use the inequalities 1-1/x <= ln(x) <= x-1 # # The original Python cod used lexical order (having converted to strings) for (num < den)) # so the results would be different e.g. for num = 9m den=200; Can this happen? What is the correct way? adj = self.exponent + number_of_digits - 1 if adj >= 1 # argument >= 10; we use 23/10 = 2.3 as a lower bound for ln(10) return _number_of_digits(adj*23/10) - 1 end if adj <= -2 # argument <= 0.1 return _number_of_digits((-1-adj)*23/10) - 1 end c = self.coefficient e = self.exponent if adj == 0 # 1 < self < 10 num = c-(10**-e) den = c return _number_of_digits(num) - _number_of_digits(den) - ((num < den) ? 1 : 0) end # adj == -1, 0.1 <= self < 1 return e + _number_of_digits(10**-e - c) - 1 end |
#_log10_exp_bound ⇒ Object
Compute a lower bound for the adjusted exponent of self.log10() In other words, find r such that self.log10() >= 10**r. Assumes that self is finite and positive and that self != 1.
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# File 'lib/flt/dec_num.rb', line 759 def _log10_exp_bound # For x >= 10 or x < 0.1 we only need a bound on the integer # part of log10(self), and this comes directly from the # exponent of x. For 0.1 <= x <= 10 we use the inequalities # 1-1/x <= log(x) <= x-1. If x > 1 we have |log10(x)| > # (1-1/x)/2.31 > 0. If x < 1 then |log10(x)| > (1-x)/2.31 > 0 # # The original Python cod used lexical order (having converted to strings) for (num < den) and (num < 231) # so the results would be different e.g. for num = 9; Can this happen? What is the correct way? adj = self.exponent + number_of_digits - 1 return _number_of_digits(adj) - 1 if adj >= 1 # self >= 10 return _number_of_digits(-1-adj)-1 if adj <= -2 # self < 0.1 c = self.coefficient e = self.exponent if adj == 0 # 1 < self < 10 num = (c - DecNum.int_radix_power(-e)) den = (231*c) return _number_of_digits(num) - _number_of_digits(den) - ((num < den) ? 1 : 0) + 2 end # adj == -1, 0.1 <= self < 1 num = (DecNum.int_radix_power(-e)-c) return _number_of_digits(num.to_i) + e - ((num < 231) ? 1 : 0) - 1 end |
#_power_exact(other, p) ⇒ Object
Attempt to compute self**other exactly Given Decimals self and other and an integer p, attempt to compute an exact result for the power self**other, with p digits of precision. Return nil if self**other is not exactly representable in p digits.
Assumes that elimination of special cases has already been performed: self and other must both be nonspecial; self must be positive and not numerically equal to 1; other must be nonzero. For efficiency, other.exponent should not be too large, so that 10**other.exponent.abs is a feasible calculation.
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# File 'lib/flt/dec_num.rb', line 558 def _power_exact(other, p) # In the comments below, we write x for the value of self and # y for the value of other. Write x = xc*10**xe and y = # yc*10**ye. # The main purpose of this method is to identify the *failure* # of x**y to be exactly representable with as little effort as # possible. So we look for cheap and easy tests that # eliminate the possibility of x**y being exact. Only if all # these tests are passed do we go on to actually compute x**y. # Here's the main idea. First normalize both x and y. We # express y as a rational m/n, with m and n relatively prime # and n>0. Then for x**y to be exactly representable (at # *any* precision), xc must be the nth power of a positive # integer and xe must be divisible by n. If m is negative # then additionally xc must be a power of either 2 or 5, hence # a power of 2**n or 5**n. # # There's a limit to how small |y| can be: if y=m/n as above # then: # # (1) if xc != 1 then for the result to be representable we # need xc**(1/n) >= 2, and hence also xc**|y| >= 2. So # if |y| <= 1/nbits(xc) then xc < 2**nbits(xc) <= # 2**(1/|y|), hence xc**|y| < 2 and the result is not # representable. # # (2) if xe != 0, |xe|*(1/n) >= 1, so |xe|*|y| >= 1. Hence if # |y| < 1/|xe| then the result is not representable. # # Note that since x is not equal to 1, at least one of (1) and # (2) must apply. Now |y| < 1/nbits(xc) iff |yc|*nbits(xc) < # 10**-ye iff len(str(|yc|*nbits(xc)) <= -ye. # # There's also a limit to how large y can be, at least if it's # positive: the normalized result will have coefficient xc**y, # so if it's representable then xc**y < 10**p, and y < # p/log10(xc). Hence if y*log10(xc) >= p then the result is # not exactly representable. # if len(str(abs(yc*xe)) <= -ye then abs(yc*xe) < 10**-ye, # so |y| < 1/xe and the result is not representable. # Similarly, len(str(abs(yc)*xc_bits)) <= -ye implies |y| # < 1/nbits(xc). xc = self.coefficient xe = self.exponent while (xc % DecNum.radix) == 0 xc /= DecNum.radix xe += 1 end yc = other.coefficient ye = other.exponent while (yc % DecNum.radix) == 0 yc /= DecNum.radix ye += 1 end # case where xc == 1: result is 10**(xe*y), with xe*y # required to be an integer if xc == 1 if ye >= 0 exponent = xe*yc*DecNum.int_radix_power(ye) else exponent, remainder = (xe*yc).divmod(DecNum.int_radix_power(-ye)) return nil if remainder!=0 end exponent = -exponent if other.sign == -1 # if other is a nonnegative integer, use ideal exponent if other.integral? and (other.sign == +1) ideal_exponent = self.exponent*other.to_i zeros = [exponent-ideal_exponent, p-1].min else zeros = 0 end return Num(+1, DecNum.int_radix_power(zeros), exponent-zeros) end # case where y is negative: xc must be either a power # of 2 or a power of 5. if other.sign == -1 last_digit = (xc % 10) if [2,4,6,8].include?(last_digit) # quick test for power of 2 return nil if xc & -xc != xc # now xc is a power of 2; e is its exponent e = _nbits(xc)-1 # find e*y and xe*y; both must be integers if ye >= 0 y_as_int = yc*DecNum.int_radix_power(ye) e = e*y_as_int xe = xe*y_as_int else ten_pow = DecNum.int_radix_power(-ye) e, remainder = (e*yc).divmod(ten_pow) return nil if remainder!=0 xe, remainder = (xe*yc).divmod(ten_pow) return nil if remainder!=0 end return nil if e*65 >= p*93 # 93/65 > log(10)/log(5) xc = 5**e elsif last_digit == 5 # e >= log_5(xc) if xc is a power of 5; we have # equality all the way up to xc=5**2658 e = _nbits(xc)*28/65 xc, remainder = (5**e).divmod(xc) return nil if remainder!=0 while (xc % 5) == 0 xc /= 5 e -= 1 end if ye >= 0 y_as_integer = DecNum.int_mult_radix_power(yc,ye) e = e*y_as_integer xe = xe*y_as_integer else ten_pow = DecNum.int_radix_power(-ye) e, remainder = (e*yc).divmod(ten_pow) return nil if remainder xe, remainder = (xe*yc).divmod(ten_pow) return nil if remainder end return nil if e*3 >= p*10 # 10/3 > log(10)/log(2) xc = 2**e else return nil end return nil if xc >= DecNum.int_radix_power(p) xe = -e-xe return Num(+1, xc, xe) end # now y is positive; find m and n such that y = m/n if ye >= 0 m, n = yc*10**ye, 1 else return nil if (xe != 0) and (_number_of_digits((yc*xe).abs) <= -ye) xc_bits = _nbits(xc) return nil if (xc != 1) and (_number_of_digits(yc.abs*xc_bits) <= -ye) m, n = yc, DecNum.int_radix_power(-ye) while ((m % 2) == 0) && ((n % 2) == 0) m /= 2 n /= 2 end while ((m % 5) == 0) && ((n % 5) == 0) m /= 5 n /= 5 end end # compute nth root of xc*10**xe if n > 1 # if 1 < xc < 2**n then xc isn't an nth power return nil if xc != 1 and xc_bits <= n xe, rem = xe.divmod(n) return nil if rem != 0 # compute nth root of xc using Newton's method a = 1 << -(-_nbits(xc)/n) # initial estimate q = r = nil loop do q, r = xc.divmod(a**(n-1)) break if a <= q a = (a*(n-1) + q)/n end return nil if !((a == q) and (r == 0)) xc = a end # now xc*10**xe is the nth root of the original xc*10**xe # compute mth power of xc*10**xe # if m > p*100/_log10_lb(xc) then m > p/log10(xc), hence xc**m > # 10**p and the result is not representable. return nil if (xc > 1) and (m > p*100/_log10_lb(xc)) xc = xc**m xe *= m return nil if xc > 10**p # by this point the result *is* exactly representable # adjust the exponent to get as close as possible to the ideal # exponent, if necessary if other.integral? && other.sign == +1 ideal_exponent = self.exponent*other.to_i zeros = [xe-ideal_exponent, p-_number_of_digits(xc)].min else zeros = 0 end return Num(+1, DecNum.int_mult_radix_power(xc, zeros), xe-zeros) end |
#_power_modulo(other, modulo, context = nil) ⇒ Object
Power-modulo: self._power_modulo(other, modulo) == (self**other) % modulo This is equivalent to Python’s 3-argument version of pow()
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# File 'lib/flt/dec_num.rb', line 499 def _power_modulo(other, modulo, context=nil) context = DecNum.define_context(context) other = _convert(other) modulo = _convert(third) if self.nan? || other.nan? || modulo.nan? return context.exception(InvalidOperation, 'sNaN', self) if self.snan? return context.exception(InvalidOperation, 'sNaN', other) if other.snan? return context.exception(InvalidOperation, 'sNaN', modulo) if other.modulo? return self._fix_nan(context) if self.nan? return other._fix_nan(context) if other.nan? return modulo._fix_nan(context) # if modulo.nan? end if !(self.integral? && other.integral? && modulo.integral?) return context.exception(InvalidOperation, '3-argument power not allowed unless all arguments are integers.') end if other < 0 return context.exception(InvalidOperation, '3-argument power cannot have a negative 2nd argument.') end if modulo.zero? return context.exception(InvalidOperation, '3-argument power cannot have a 0 3rd argument.') end if modulo.adjusted_exponent >= context.precision return context.exception(InvalidOperation, 'insufficient precision: power 3rd argument must not have more than precision digits') end if other.zero? && self.zero? return context.exception(InvalidOperation, "0**0 not defined") end sign = other.even? ? +1 : -1 modulo = modulo.to_i.abs base = (self.coefficient % modulo * (DecNum.int_radix_power(self.exponent) % modulo)) % modulo other.exponent.times do base = (base**DecNum.radix) % modulo end base = (base**other.coefficient) % modulo Num(sign, base, 0) end |
#exp(context = nil) ⇒ Object
Exponential function
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# File 'lib/flt/dec_num.rb', line 377 def exp(context=nil) context = DecNum.define_context(context) # exp(NaN) = NaN ans = _check_nans(context) return ans if ans # exp(-Infinity) = 0 return DecNum.zero if self.infinite? && (self.sign == -1) # exp(0) = 1 return Num(1) if self.zero? # exp(Infinity) = Infinity return Num(self) if self.infinite? # the result is now guaranteed to be inexact (the true # mathematical result is transcendental). There's no need to # raise Rounded and Inexact here---they'll always be raised as # a result of the call to _fix. return context.exception(Inexact, 'Inexact exp') if context.exact? p = context.precision adj = self.adjusted_exponent # we only need to do any computation for quite a small range # of adjusted exponents---for example, -29 <= adj <= 10 for # the default context. For smaller exponent the result is # indistinguishable from 1 at the given precision, while for # larger exponent the result either overflows or underflows. if self.sign == +1 and adj > _number_of_digits((context.emax+1)*3) # overflow ans = Num(+1, 1, context.emax+1) elsif self.sign == -1 and adj > _number_of_digits((-context.etiny+1)*3) # underflow to 0 ans = Num(+1, 1, context.etiny-1) elsif self.sign == +1 and adj < -p # p+1 digits; final round will raise correct flags ans = Num(+1, DecNum.int_radix_power(p)+1, -p) elsif self.sign == -1 and adj < -p-1 # p+1 digits; final round will raise correct flags ans = Num(+1, DecNum.int_radix_power(p+1)-1, -p-1) else # general case c = self.coefficient e = self.exponent c = -c if self.sign == -1 # compute correctly rounded result: increase precision by # 3 digits at a time until we get an unambiguously # roundable result extra = 3 coeff = exp = nil loop do coeff, exp = _dexp(c, e, p+extra) break if (coeff % (5*10**(_number_of_digits(coeff)-p-1)))!=0 extra += 3 end ans = Num(+1, coeff, exp) end # at this stage, ans should round correctly with *any* # rounding mode, not just with ROUND_HALF_EVEN DecNum.context(context, :rounding=>:half_even) do |local_context| ans = ans._fix(local_context) context.flags = local_context.flags end return ans end |
#ln(context = nil) ⇒ Object
Returns the natural (base e) logarithm
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# File 'lib/flt/dec_num.rb', line 448 def ln(context=nil) context = DecNum.define_context(context) # ln(NaN) = NaN ans = _check_nans(context) return ans if ans # ln(0.0) == -Infinity return DecNum.infinity(-1) if self.zero? # ln(Infinity) = Infinity return DecNum.infinity if self.infinite? && self.sign == +1 # ln(1.0) == 0.0 return DecNum.zero if self == Num(1) # ln(negative) raises InvalidOperation return context.exception(InvalidOperation, 'ln of a negative value') if self.sign==-1 # result is irrational, so necessarily inexact return context.exception(Inexact, 'Inexact exp') if context.exact? c = self.coefficient e = self.exponent p = context.precision # correctly rounded result: repeatedly increase precision by 3 # until we get an unambiguously roundable result places = p - self._ln_exp_bound + 2 # at least p+3 places coeff = nil loop do coeff = _dlog(c, e, places) # assert coeff.to_s.length-p >= 1 break if (coeff % (5*10**(_number_of_digits(coeff.abs)-p-1))) != 0 places += 3 end ans = Num((coeff<0) ? -1 : +1, coeff.abs, -places) DecNum.context(context, :rounding=>:half_even) do |local_context| ans = ans._fix(local_context) context.flags = local_context.flags end return ans end |
#log10(context = nil) ⇒ Object
Returns the base 10 logarithm
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# File 'lib/flt/dec_num.rb', line 327 def log10(context=nil) context = DecNum.define_context(context) # log10(NaN) = NaN ans = _check_nans(context) return ans if ans # log10(0.0) == -Infinity return DecNum.infinity(-1) if self.zero? # log10(Infinity) = Infinity return DecNum.infinity if self.infinite? && self.sign == +1 # log10(negative or -Infinity) raises InvalidOperation return context.exception(InvalidOperation, 'log10 of a negative value') if self.sign == -1 digits = self.digits # log10(10**n) = n if digits.first == 1 && digits[1..-1].all?{|d| d==0} # answer may need rounding ans = Num(self.exponent + digits.size - 1) return ans if context.exact? else # result is irrational, so necessarily inexact return context.exception(Inexact, "Inexact power") if context.exact? c = self.coefficient e = self.exponent p = context.precision # correctly rounded result: repeatedly increase precision # until result is unambiguously roundable places = p-self._log10_exp_bound+2 coeff = nil loop do coeff = _dlog10(c, e, places) # assert coeff.abs.to_s.length-p >= 1 break if (coeff % (5*10**(_number_of_digits(coeff.abs)-p-1)))!=0 places += 3 end ans = Num(coeff<0 ? -1 : +1, coeff.abs, -places) end DecNum.context(context, :rounding=>:half_even) do |local_context| ans = ans._fix(local_context) context.flags = local_context.flags end return ans end |
#number_of_digits ⇒ Object
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# File 'lib/flt/dec_num.rb', line 119 def number_of_digits @coeff.is_a?(Integer) ? _number_of_digits(@coeff) : 0 end |
#power(other, modulo = nil, context = nil) ⇒ Object
Raises to the power of x, to modulo if given.
With two arguments, compute self**other. If self is negative then other must be integral. The result will be inexact unless other is integral and the result is finite and can be expressed exactly in ‘precision’ digits.
With three arguments, compute (self**other) % modulo. For the three argument form, the following restrictions on the arguments hold:
- all three arguments must be integral
- other must be nonnegative
- at least one of self or other must be nonzero
- modulo must be nonzero and have at most 'precision' digits
The result of a.power(b, modulo) is identical to the result that would be obtained by computing (a**b) % modulo with unbounded precision, but is computed more efficiently. It is always exact.
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# File 'lib/flt/dec_num.rb', line 143 def power(other, modulo=nil, context=nil) if context.nil? && (modulo.is_a?(Context) || modulo.is_a?(Hash)) context = modulo modulo = nil end return self.power_modulo(other, modulo, context) if modulo context = DecNum.define_context(context) other = _convert(other) ans = _check_nans(context, other) return ans if ans # 0**0 = NaN (!), x**0 = 1 for nonzero x (including +/-Infinity) if other.zero? if self.zero? return context.exception(InvalidOperation, '0 ** 0') else return Num(1) end end # result has sign -1 iff self.sign is -1 and other is an odd integer result_sign = +1 _self = self if _self.sign == -1 if other.integral? result_sign = -1 if !other.even? else # -ve**noninteger = NaN # (-0)**noninteger = 0**noninteger unless self.zero? return context.exception(InvalidOperation, 'x ** y with x negative and y not an integer') end end # negate self, without doing any unwanted rounding _self = self.copy_negate end # 0**(+ve or Inf)= 0; 0**(-ve or -Inf) = Infinity if _self.zero? return (other.sign == +1) ? Num(result_sign, 0, 0) : num_class.infinity(result_sign) end # Inf**(+ve or Inf) = Inf; Inf**(-ve or -Inf) = 0 if _self.infinite? return (other.sign == +1) ? num_class.infinity(result_sign) : Num(result_sign, 0, 0) end # 1**other = 1, but the choice of exponent and the flags # depend on the exponent of self, and on whether other is a # positive integer, a negative integer, or neither if _self == Num(1) return _self if context.exact? if other.integral? # exp = max(self._exp*max(int(other), 0), # 1-context.prec) but evaluating int(other) directly # is dangerous until we know other is small (other # could be 1e999999999) if other.sign == -1 multiplier = 0 elsif other > context.precision multiplier = context.precision else multiplier = other.to_i end exp = _self.exponent * multiplier if exp < 1-context.precision exp = 1-context.precision context.exception Rounded end else context.exception Rounded context.exception Inexact exp = 1-context.precision end return Num(result_sign, DecNum.int_radix_power(-exp), exp) end # compute adjusted exponent of self self_adj = _self.adjusted_exponent # self ** infinity is infinity if self > 1, 0 if self < 1 # self ** -infinity is infinity if self < 1, 0 if self > 1 if other.infinite? if (other.sign == +1) == (self_adj < 0) return Num(result_sign, 0, 0) else return DecNum.infinity(result_sign) end end # from here on, the result always goes through the call # to _fix at the end of this function. ans = nil # crude test to catch cases of extreme overflow/underflow. If # log10(self)*other >= 10**bound and bound >= len(str(Emax)) # then 10**bound >= 10**len(str(Emax)) >= Emax+1 and hence # self**other >= 10**(Emax+1), so overflow occurs. The test # for underflow is similar. bound = _self._log10_exp_bound + other.adjusted_exponent if (self_adj >= 0) == (other.sign == +1) # self > 1 and other +ve, or self < 1 and other -ve # possibility of overflow if bound >= _number_of_digits(context.emax) ans = Num(result_sign, 1, context.emax+1) end else # self > 1 and other -ve, or self < 1 and other +ve # possibility of underflow to 0 etiny = context.etiny if bound >= _number_of_digits(-etiny) ans = Num(result_sign, 1, etiny-1) end end # try for an exact result with precision +1 if ans.nil? if context.exact? if other.adjusted_exponent < 100 test_precision = _self.number_of_digits*other.to_i+1 else test_precision = _self.number_of_digits+1 end else test_precision = context.precision + 1 end ans = _self._power_exact(other, test_precision) if !ans.nil? && (result_sign == -1) ans = Num(-1, ans.coefficient, ans.exponent) end end # usual case: inexact result, x**y computed directly as exp(y*log(x)) if !ans.nil? return ans if context.exact? else return context.exception(Inexact, "Inexact power") if context.exact? p = context.precision xc = _self.coefficient xe = _self.exponent yc = other.coefficient ye = other.exponent yc = -yc if other.sign == -1 # compute correctly rounded result: start with precision +3, # then increase precision until result is unambiguously roundable extra = 3 coeff, exp = nil, nil loop do coeff, exp = _dpower(xc, xe, yc, ye, p+extra) #break if (coeff % DecNum.int_mult_radix_power(5,coeff.to_s.length-p-1)) != 0 break if (coeff % (5*10**(_number_of_digits(coeff)-p-1))) != 0 extra += 3 end ans = Num(result_sign, coeff, exp) end # the specification says that for non-integer other we need to # raise Inexact, even when the result is actually exact. In # the same way, we need to raise Underflow here if the result # is subnormal. (The call to _fix will take care of raising # Rounded and Subnormal, as usual.) if !other.integral? context.exception Inexact # pad with zeros up to length context.precision+1 if necessary if ans.number_of_digits <= context.precision expdiff = context.precision+1 - ans.number_of_digits ans = Num(ans.sign, DecNum.int_mult_radix_power(ans.coefficient, expdiff), ans.exponent-expdiff) end context.exception Underflow if ans.adjusted_exponent < context.emin end # unlike exp, ln and log10, the power function respects the # rounding mode; no need to use ROUND_HALF_EVEN here ans._fix(context) end |