Merge pull request #16480 from JuliaLang/sb/rational

Fix rationalize for small values

base/rational.jl

@@ -111,11 +111,14 @@ function rationalize{T<:Integer}(::Type{T}, x::AbstractFloat; tol::Real=eps(x))
     r = x-a
     y = one(x)
 
-    nt, t, tt = tol, zero(tol), zero(tol)
+    nt, t, tt = tol, zero(tol), tol
 
+    # compute the successive convergents of the continued fraction
+    #  np // nq = (p*a + pp) // (q*a + qq)
     while r > nt
         try
             ia = convert(T,a)
+
             np = checked_add(checked_mul(ia,p),pp)
             nq = checked_add(checked_mul(ia,q),qq)
             p, pp = np, p
@@ -125,16 +128,21 @@ function rationalize{T<:Integer}(::Type{T}, x::AbstractFloat; tol::Real=eps(x))
             return p // q
         end
 
-        t, tt = nt, t
+        # naive approach of using
+        #   x = 1/r; a = trunc(x); r = x - a
+        # is inexact, so we store x as x/y
         x, y = y, r
-
         a, r = divrem(x,y)
-        nt = a*t + tt
+
+        # maintain
+        # x0 = (p + (-1)^i * r) / q
+        t, tt = nt, t
+        nt = a*t+tt
     end
 
     # find optimal semiconvergent
     # smallest a such that x-a*y < a*t+tt
-    a = cld(x-tt,y+t)
+    a = max(cld(x-tt,y+t),0.0)
     try
         ia = convert(T,a)
         np = checked_add(checked_mul(ia,p),pp)

test/numbers.jl

@@ -2131,6 +2131,10 @@ for f in (trunc, round, floor, ceil)
          103993//33102
          312689//99532 ]
 
+# issue 16311
+rationalize(nextfloat(0.0)) == 0//1
+
+
 # primes
 
 @test primes(10000) == primes(2, 10000) == [