# Class: Flt::Support::Rationalizer

Inherits:
Object
• Object
show all
Extended by:
AuxiliarFunctions
Includes:
AuxiliarFunctions
Defined in:
lib/flt/support/rationalizer.rb,
lib/flt/support/rationalizer_extra.rb

## Overview

This class provides efficient conversion of fraction (as approximate floating point numbers) to rational numbers.

## Defined Under Namespace

Modules: AuxiliarFunctions

## Class Method Summary collapse

• Best fraction given maximum denominator Algorithm Copyright © 1991 by Joseph K.

• Exact conversion to rational.

## Instance Method Summary collapse

• An exact rationalization method for binary floating point that yields smallest fractions when possible and is not too slow.

• An a here’s a shorter implementation relying on the semantics of the power operator, but which is somewhat slow:.

• constructor

Create Rationalizator with given tolerance.

• Rationalization method that finds the fraction with smallest denominator fraction within the tolerance distance of an approximate (floating point) number.

• This is algorithm PDQ2 by Joe Horn.

• Smallest denominator rationalization procedure by Joe Horn and Tony Hutchins; this is the most efficient method as implemented in RPL.

• Simple Rationalization by Joe Horn.

• This is from a RPL program by Tony Hutchins (PDR6).

• This algorithm is derived from exercise 39 of 4.5.3 in “The Art of Computer Programming”, by Donald E.

• Smallest denominator rationalization based on exercise 39 of cite[S 4.5.3]Knuth.

• La siguiente variante realiza una iteración menos si xq<xp y una iteración más si xq>xp.

## Constructor Details

### #initialize(tol = Tolerance(:epsilon)) ⇒ Rationalizer

Create Rationalizator with given tolerance.

 ``` 113 114 115``` ```# File 'lib/flt/support/rationalizer.rb', line 113 def initialize(tol=Tolerance(:epsilon)) @tol = tol end```

## Class Method Details

### .[](*args) ⇒ Object

 ``` 117 118 119``` ```# File 'lib/flt/support/rationalizer.rb', line 117 def self.[](*args) new(*args) end```

### .max_denominator(f, max_den = 1000000000, num_class = nil) ⇒ Object

Best fraction given maximum denominator Algorithm Copyright © 1991 by Joseph K. Horn.

The implementation of this method uses floating point arithmetic which limits the magnitude and precision of the results, specially using Float values.

 ``` 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276``` ```# File 'lib/flt/support/rationalizer.rb', line 242 def self.max_denominator(f, max_den=1000000000, num_class=nil) return rationalize_special(f) if special?(f) return nil if max_den < 1 num_class ||= f.class context = num_class.context return ip(f),1 if fp(f) == 0 cast = lambda{|x| context.Num(x)} one = cast[1] sign = f < 0 f = -f if sign a,b,c = 0,1,f while b < max_den && c != 0 cc = one/c a,b,c = b, ip(cc)*b+a, fp(cc) end if b>max_den b -= a*ceil(cast[b-max_den]/a) end f1,f2 = [a,b].collect{|x| abs(cast[rnd(x*f)]/x-f)} a = f1 > f2 ? b : a num,den = rnd(a*f).to_i,a den = 1 if abs(den) < 1 num = -num if sign return num,den end```

### .rationalize_special(x) ⇒ Object

 ``` 278 279 280 281 282 283 284``` ```# File 'lib/flt/support/rationalizer.rb', line 278 def self.rationalize_special(x) if x.nan? [0, 0] else [sign(x), 0] end end```

### .to_r(x) ⇒ Object

Exact conversion to rational. Ruby provides this method for all numeric types since version 1.9.1, but before that it wasn’t available for Float or BigDecimal. This methods supports old Ruby versions.

 ``` 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57``` ```# File 'lib/flt/support/rationalizer.rb', line 10 def self.to_r(x) if x.respond_to?(:to_r) x.to_r else case x when Float # Float did not had a #to_r method until Ruby 1.9.1 return Rational(x.to_i, 1) if x.modulo(1) == 0 if !x.finite? return Rational(0, 0) if x.nan? return x < 0 ? Rational(-1, 0) : Rational(1, 0) end f, e = Math.frexp(x) if e < Float::MIN_EXP bits = e + Float::MANT_DIG - Float::MIN_EXP else bits = [Float::MANT_DIG,e].max # return Rational(x.to_i, 1) if bits < e end p = Math.ldexp(f, bits) e = bits - e if e < Float::MAX_EXP q = Math.ldexp(1, e) else q = Float::RADIX**e end Rational(p.to_i, q.to_i) when BigDecimal # BigDecimal probably didn't have #to_r at some point s, f, b, e = x.split p = f.to_i p = -p if s < 0 e = f.size - e if e < 0 p *= b**(-e) e = 0 end q = b**(e) Rational(p,q) else x.to_r end end end```

## Instance Method Details

### #exact_binary_rationalization(x) ⇒ Object

An exact rationalization method for binary floating point that yields smallest fractions when possible and is not too slow

 ``` 147 148 149 150 151 152 153 154``` ```# File 'lib/flt/support/rationalizer_extra.rb', line 147 def exact_binary_rationalization(x) p, q = x, 1 while p.modulo(1) != 0 p *= 2.0 q <<= 1 # q *= 2 end Rational(p.to_i, q) end```

### #exact_float_rationalization(x) ⇒ Object

An a here’s a shorter implementation relying on the semantics of the power operator, but which is somewhat slow:

 ``` 158 159 160 161 162 163``` ```# File 'lib/flt/support/rationalizer_extra.rb', line 158 def exact_float_rationalization(x) f,e = Math.frexp(x) f = Math.ldexp(f, Float::MANT_DIG) e -= Float::MANT_DIG return Rational(f.to_i*(Float::RADIX**e.to_i), 1) end```

### #rationalize(x) ⇒ Object

Rationalization method that finds the fraction with smallest denominator fraction within the tolerance distance of an approximate (floating point) number.

 ``` 125 126 127 128``` ```# File 'lib/flt/support/rationalizer.rb', line 125 def rationalize(x) # Use the algorithm which has been found most efficient, rationalize_Knuth. rationalize_Knuth(x) end```

### #rationalize_Horn(x) ⇒ Object

This is algorithm PDQ2 by Joe Horn.

 ``` 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198``` ```# File 'lib/flt/support/rationalizer.rb', line 165 def rationalize_Horn(x) rationalization(x) do |z, t| a,b = num_den(t) n0,d0 = (n,d = num_den(z)) cn,x,pn,cd,y,pd,lo,hi,mid,q,r = 1,1,0,0,0,1,0,1,1,0,0 begin q,r = n.divmod(d) x = q*cn+pn y = q*cd+pd pn = cn cn = x pd = cd cd = y n,d = d,r end until b*(n0*y-d0*x).abs <= a*d0*y if q > 1 hi = q begin mid = (lo + hi).div(2) x = cn - pn*mid y = cd - pd*mid if b*(n0*y - d0*x).abs <= a*d0*y lo = mid else hi = mid end end until hi - lo <= 1 x = cn - pn*lo y = cd - pd*lo end [x, y] end end```

### #rationalize_Horn_Hutchins(value) ⇒ Object

Smallest denominator rationalization procedure by Joe Horn and Tony Hutchins; this is the most efficient method as implemented in RPL. Tony Hutchins has come up with PDR6, an improvement over PDQ2; though benchmarking does not show any speed improvement under Ruby.

 ``` 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66``` ```# File 'lib/flt/support/rationalizer_extra.rb', line 50 def rationalize_Horn_Hutchins(value) rationalization(value) do |x, dx| a,b = num_den(dx) n,d = num_den(x) pc,ce = n,-d pc,cd = 1,0 t = a*b begin tt = (-pe).div(ce) pd,cd = cd,pd+tt*cd pe,ce = ce,pe+tt*ce end until b*ce.abs <= t*cd tt = t * (pe<0 ? -1 : (pe>0 ? +1 : 0)) tt = (tt*d+b*ce).div(tt*pd+b*pe) [(n*cd-ce-(n*pd-pe)*tt)/d, tt/(cd-tt*pd)] end end```

### #rationalize_Horn_simple(x, smallest_denominator = false) ⇒ Object

Simple Rationalization by Joe Horn

 ``` 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43``` ```# File 'lib/flt/support/rationalizer_extra.rb', line 9 def rationalize_Horn_simple(x, smallest_denominator = false) rationalization(x) do |z, t| a,b = num_den(t) n0,d0 = (n,d = z.nio_xr.nio_num_den) cn,x,pn,cd,y,pd,lo,hi,mid,q,r = 1,1,0,0,0,1,0,1,1,0,0 begin q,r = n.divmod(d) x = q*cn+pn y = q*cd+pd pn = cn cn = x pd = cd cd = y n,d = d,r end until b*(n0*y-d0*x).abs <= a*d0*y if smallest_denominator if q>1 hi = q begin mid = (lo+hi).div(2) x = cn-pn*mid y = cd-pd*mid if b*(n0*y-d0*x).abs <= a*d0*y lo = mid else hi = mid end end until hi-lo <= 1 x = cn - pn*lo y = cd - pd*lo end end [x, y] end end```

### #rationalize_HornHutchins(x) ⇒ Object

This is from a RPL program by Tony Hutchins (PDR6).

 ``` 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234``` ```# File 'lib/flt/support/rationalizer.rb', line 201 def rationalize_HornHutchins(x) rationalization(x) do |z, t| a,b = num_den(t) n0,d0 = (n,d = num_den(z)) cn,x,pn,cd,y,pd,lo,hi,mid,q,r = 1,1,0,0,0,1,0,1,1,0,0 begin q,r = n.divmod(d) x = q*cn+pn y = q*cd+pd pn = cn cn = x pd = cd cd = y n,d = d,r end until b*(n0*y-d0*x).abs <= a*d0*y if q > 1 hi = q begin mid = (lo + hi).div(2) x = cn - pn*mid y = cd - pd*mid if b*(n0*y - d0*x).abs <= a*d0*y lo = mid else hi = mid end end until hi - lo <= 1 x = cn - pn*lo y = cd - pd*lo end [x, y] end end```

### #rationalize_Knuth(value) ⇒ Object

This algorithm is derived from exercise 39 of 4.5.3 in “The Art of Computer Programming”, by Donald E. Knuth.

 ``` 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162``` ```# File 'lib/flt/support/rationalizer.rb', line 132 def rationalize_Knuth(value) rationalization(value) do |x, dx| x = to_r(x) dx = to_r(dx) xp,xq = num_den(x-dx) yp,yq = num_den(x+dx) a = [] fin, odd = false, false while !fin && xp != 0 && yp != 0 odd = !odd xp,xq = xq,xp ax = xp.div(xq) xp -= ax*xq yp,yq = yq,yp ay = yp.div(yq) yp -= ay*yq if ax!=ay fin = true ax,xp,xq = ay,yp,yq if odd end a << ax # .to_i end a[-1] += 1 if xp != 0 && a.size > 0 p,q = 1,0 (1..a.size).each{|i| p, q = q+p*a[-i], p} [q, p] end end```

### #rationalize_Knuth_Goizueta(value) ⇒ Object

Smallest denominator rationalization based on exercise 39 of cite[S 4.5.3]Knuth. This has been found the most efficient method (except for large tolerances) as implemented in Ruby. Here’s the rationalization procedure based on the exercise by Knuth. We need first to calculate the limits (x-dx, x+dx) of the range where we’ll look for the rational number. If we compute them using floating point and then convert then to fractions this method is always more efficient than the other procedures implemented here, but it may be less accurate. We can achieve perfect accuracy as the other methods by doing the substraction and addition with rationals, but then this method becomes less efficient than the others for a low number of iterations (low precision required).

 ``` 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109``` ```# File 'lib/flt/support/rationalizer_extra.rb', line 79 def rationalize_Knuth_Goizueta(value) rationalization(value) do |x, dx| x = to_r(x) dx = to_r(dx) xp,xq = num_den(x-dx) yp,yq = num_den(x+dx) a = [] fin,odd = false,false while !fin && xp!=0 && yp!=0 odd = !odd xp,xq = xq,xp ax = xp.div(xq) xp -= ax*xq yp,yq = yq,yp ay = yp.div(yq) yp -= ay*yq if ax!=ay fin = true ax,xp,xq = ay,yp,yq if odd end a << ax # .to_i end a[-1] += 1 if xp!=0 && a.size>0 p,q = 1,0 (1..a.size).each{|i| p,q=q+p*a[-i],p} [q, p] end end```

### #rationalize_Knuth_Goizueta_b(value) ⇒ Object

La siguiente variante realiza una iteración menos si xq<xp y una iteración más si xq>xp.

 ``` 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143``` ```# File 'lib/flt/support/rationalizer_extra.rb', line 113 def rationalize_Knuth_Goizueta_b(value) rationalization(value) do |x, dx| x = to_r(x) dx = to_r(dx) xq,xp = num_den(x-dx) yq,yp = num_den(x+dx) a = [] fin,odd = false,false while !fin && xp!=0 && yp!=0 odd = !odd xp,xq = xq,xp ax = xp.div(xq) xp -= ax*xq yp,yq = yq,yp ay = yp.div(yq) yp -= ay*yq if ax!=ay fin = true ax,xp,xq = ay,yp,yq if odd end a << ax # .to_i end a[-1] += 1 if xp!=0 && a.size>0 p,q = 1,0 (1..a.size).each{|i| p,q=q+p*a[-i],p} [p, q] end end```