diff --git a/NEWS.md b/NEWS.md
index 978838ee128a6..88e952d3ad429 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -19,6 +19,9 @@ Language changes
   * Declaring arguments as `x::ANY` to avoid specialization has been replaced
     by `@nospecialize x`. ([#22666]).
 
+  * The parsing of `1<<2*3` as `1<<(2*3)` is deprecated, and will change to
+    `(1<<2)*3` in a future version ([#13079]).
+
 Breaking changes
 ----------------
 
diff --git a/base/deprecated.jl b/base/deprecated.jl
index 51d9cd4e63fff..6d6cb80788759 100644
--- a/base/deprecated.jl
+++ b/base/deprecated.jl
@@ -1563,6 +1563,12 @@ end
 # ::ANY is deprecated in src/method.c
 # also remove all instances of `jl_ANY_flag` in src/
 
+# issue #13079
+# in julia-parser.scm:
+#     move prec-bitshift after prec-rational
+#     remove parse-with-chains-warn and bitshift-warn
+# update precedence table in doc/src/manual/mathematical-operations.md
+
 # PR #22182
 @deprecate is_apple   Sys.isapple
 @deprecate is_bsd     Sys.isbsd
diff --git a/base/dict.jl b/base/dict.jl
index e78a99a32d30b..e0ddcaaee649e 100644
--- a/base/dict.jl
+++ b/base/dict.jl
@@ -271,7 +271,7 @@ function rehash!(h::Dict{K,V}, newsz = length(h.keys)) where V where K
 end
 
 max_values(::Type) = typemax(Int)
-max_values(T::Type{<:Union{Void,BitIntegerSmall}}) = 1 << 8*sizeof(T)
+max_values(T::Type{<:Union{Void,BitIntegerSmall}}) = 1 << (8*sizeof(T))
 max_values(T::Union) = max(max_values(T.a), max_values(T.b))
 max_values(::Type{Bool}) = 2
 
diff --git a/base/grisu/fastshortest.jl b/base/grisu/fastshortest.jl
index f9009a7703753..c7b70dea84acc 100644
--- a/base/grisu/fastshortest.jl
+++ b/base/grisu/fastshortest.jl
@@ -55,7 +55,7 @@ const SmallPowersOfTen = [
         1000000, 10000000, 100000000, 1000000000]
 
 function bigpowten(n,n_bits)
-    guess = (n_bits + 1) * 1233 >> 12
+    guess = ((n_bits + 1) * 1233) >> 12
     guess += 1
     i = SmallPowersOfTen[guess+1]
     return n < i ? (SmallPowersOfTen[guess], guess-1) : (i,guess)
diff --git a/base/multinverses.jl b/base/multinverses.jl
index d436860f14506..319072e7214ac 100644
--- a/base/multinverses.jl
+++ b/base/multinverses.jl
@@ -135,12 +135,12 @@ end
 UnsignedMultiplicativeInverse(x::Unsigned) = UnsignedMultiplicativeInverse{typeof(x)}(x)
 
 function div(a::T, b::SignedMultiplicativeInverse{T}) where T
-    x = ((widen(a)*b.multiplier) >>> sizeof(a)*8) % T
+    x = ((widen(a)*b.multiplier) >>> (sizeof(a)*8)) % T
     x += (a*b.addmul) % T
     ifelse(abs(b.divisor) == 1, a*b.divisor, (signbit(x) + (x >> b.shift)) % T)
 end
 function div(a::T, b::UnsignedMultiplicativeInverse{T}) where T
-    x = ((widen(a)*b.multiplier) >>> sizeof(a)*8) % T
+    x = ((widen(a)*b.multiplier) >>> (sizeof(a)*8)) % T
     x = ifelse(b.add, convert(T, convert(T, (convert(T, a - x) >>> 1)) + x), x)
     ifelse(b.divisor == 1, a, x >>> b.shift)
 end
diff --git a/base/random.jl b/base/random.jl
index 79c2e5cdb4987..9aa296a3e0c29 100644
--- a/base/random.jl
+++ b/base/random.jl
@@ -614,7 +614,7 @@ function rand!(r::MersenneTwister, A::Array{UInt128}, n::Int=length(A))
     if n > 0
         u = rand_ui2x52_raw(r)
         for i = 1:n
-            @inbounds A[i] ⊻= u << 12*i
+            @inbounds A[i] ⊻= u << (12*i)
         end
     end
     A
diff --git a/src/julia-parser.scm b/src/julia-parser.scm
index dc142f5b5f6ea..60d6bcd5f167b 100644
--- a/src/julia-parser.scm
+++ b/src/julia-parser.scm
@@ -865,9 +865,48 @@
                   (else
                    (loop (list 'call t ex (down s))))))))))
 
+(define (parse-with-chains-warn s down ops chain-ops)
+  (let loop ((ex (down s))
+             (got #f))
+    (let ((t (peek-token s)))
+      (if (not (ops t))
+          (cons ex got)
+          (let ((spc (ts:space? s)))
+            (take-token s)
+            (cond ((and space-sensitive spc (memq t unary-and-binary-ops)
+                        (not (eqv? (peek-char (ts:port s)) #\ )))
+                   ;; here we have "x -y"
+                   (ts:put-back! s t spc)
+                   (cons ex got))
+                  ((memq t chain-ops)
+                   (loop (list* 'call t ex
+                                (parse-chain s down t))
+                         #t))
+                  (else
+                   (loop (list 'call t ex (down s))
+                         got))))))))
+
 (define (parse-expr s)     (parse-with-chains s parse-shift         is-prec-plus? '(+ ++)))
-(define (parse-shift s)    (parse-LtoR        s parse-term          is-prec-bitshift?))
-(define (parse-term s)     (parse-with-chains s parse-rational      is-prec-times? '(*)))
+
+(define (bitshift-warn s)
+  (syntax-deprecation s (string "call to `*` inside call to bitshift operator")
+                      "parenthesized call to `*`"))
+
+(define (parse-shift s)    #;(parse-LtoR        s parse-term          is-prec-bitshift?)
+  (let loop ((ex (parse-term s))
+             (t  (peek-token s))
+             (warn1 #f))
+    (let ((ex (car ex))
+          (warn (cdr ex)))
+      (if (is-prec-bitshift? t)
+          (begin (if warn (bitshift-warn s))
+                 (take-token s)
+                 (let ((nxt (parse-term s)))
+                   (loop (cons (list 'call t ex (car nxt)) (cdr nxt)) (peek-token s) (cdr nxt))))
+          (begin (if warn1 (bitshift-warn s))
+                 ex)))))
+
+(define (parse-term s)     (parse-with-chains-warn s parse-rational      is-prec-times? '(*)))
 (define (parse-rational s) (parse-LtoR        s parse-unary-subtype is-prec-rational?))
 
 ;; parse `<: A where B` as `<: (A where B)` (issue #21545)