diff --git a/base/array.jl b/base/array.jl
index 308f31c40371c..0a24dc4fd0785 100644
--- a/base/array.jl
+++ b/base/array.jl
@@ -1087,14 +1087,14 @@ end
 # Specifically we are wasting ~10% of memory for small arrays
 # by not picking memory sizes that max out a GC pool
 function overallocation(maxsize)
-    maxsize < 8 && return 8;
-    # compute maxsize = maxsize + 4*maxsize^(7/8) + maxsize/8
+    # compute maxsize = maxsize + 3*maxsize^(7/8) + maxsize/8
     # for small n, we grow faster than O(n)
     # for large n, we grow at O(n/8)
     # and as we reach O(memory) for memory>>1MB,
     # this means we end by adding about 10% of memory each time
+    # most commonly, this will take steps of 0-3-9-34 or 1-4-16-66 or 2-8-33
     exp2 = sizeof(maxsize) * 8 - Core.Intrinsics.ctlz_int(maxsize)
-    maxsize += (1 << div(exp2 * 7, 8)) * 4 + div(maxsize, 8)
+    maxsize += (1 << div(exp2 * 7, 8)) * 3 + div(maxsize, 8)
     return maxsize
 end
 
diff --git a/base/iobuffer.jl b/base/iobuffer.jl
index 1a793ec2eef71..6645ffeeb777a 100644
--- a/base/iobuffer.jl
+++ b/base/iobuffer.jl
@@ -604,7 +604,8 @@ end
     end
     # The fast path here usually checks there is already room, then does nothing.
     # When append is true, new data is added after io.size, not io.ptr
-    existing_space = min(lastindex(io.data), io.maxsize + get_offset(io)) - (io.append ? io.size : io.ptr - 1)
+    start_offset = io.append ? io.size : io.ptr - 1
+    existing_space = min(lastindex(io.data) - start_offset, io.maxsize - (start_offset - get_offset(io)))
     if existing_space < nshort % Int
         # Outline this function to make it more likely that ensureroom inlines itself
         return ensureroom_slowpath(io, nshort, existing_space)
@@ -898,12 +899,13 @@ function unsafe_write(to::GenericIOBuffer, p::Ptr{UInt8}, nb::UInt)
     append = to.append
     ptr = append ? size+1 : to.ptr
     data = to.data
-    to_write = min(nb, (min(Int(length(data))::Int, to.maxsize + get_offset(to)) - ptr + 1) % UInt) % Int
+    start_offset = ptr - 1
+    to_write = max(0, min(nb, (min(Int(length(data))::Int - start_offset, to.maxsize - (start_offset - get_offset(to)))) % UInt) % Int)
     # Dispatch based on the type of data, to possibly allow using memcpy
     _unsafe_write(data, p, ptr, to_write % UInt)
     # Update to.size only if the ptr has advanced to higher than
     # the previous size. Otherwise, we just overwrote existing data
-    to.size = max(size, ptr + to_write - 1)
+    to.size = max(size, start_offset + to_write)
     # If to.append, we only update size, not ptr.
     if !append
         to.ptr = ptr + to_write
@@ -941,7 +943,7 @@ end
     ptr = (to.append ? to.size+1 : to.ptr)
     # We have just ensured there is room for 1 byte, EXCEPT if we were to exceed
     # maxsize. So, we just need to check that here.
-    if ptr > to.maxsize + get_offset(to)
+    if ptr - get_offset(to) > to.maxsize
         return 0
     else
         to.data[ptr] = a
diff --git a/base/stream.jl b/base/stream.jl
index 40106436ebd05..ead65d9b59706 100644
--- a/base/stream.jl
+++ b/base/stream.jl
@@ -612,7 +612,8 @@ end
 function alloc_request(buffer::IOBuffer, recommended_size::UInt)
     ensureroom(buffer, recommended_size)
     ptr = buffer.append ? buffer.size + 1 : buffer.ptr
-    nb = min(length(buffer.data), buffer.maxsize + get_offset(buffer)) - ptr + 1
+    start_offset = ptr - 1
+    nb = max(0, min(length(buffer.data) - start_offset, buffer.maxsize - (start_offset - get_offset(buffer))))
     return (Ptr{Cvoid}(pointer(buffer.data, ptr)), nb)
 end
 
diff --git a/test/arrayops.jl b/test/arrayops.jl
index d4b401152d7ef..95314c3c85fb2 100644
--- a/test/arrayops.jl
+++ b/test/arrayops.jl
@@ -525,6 +525,12 @@ end
     v = empty!(collect(1:100))
     pushfirst!(v, 1)
     @test length(v.ref.mem) == 100
+
+    # test that insert! at position 1 doesn't allocate for empty arrays with capacity (issue #58640)
+    v = empty!(Vector{Int}(undef, 5))
+    insert!(v, 1, 10)
+    @test v == [10]
+    @test length(v.ref.mem) == 5
 end
 
 @testset "popat!(::Vector, i, [default])" begin
diff --git a/test/core.jl b/test/core.jl
index 5e8acf5740f34..618189fea6bbe 100644
--- a/test/core.jl
+++ b/test/core.jl
@@ -6852,9 +6852,6 @@ Base._growat!(A, 4, 1)
 Base._growat!(A, 2, 3)
 @test getindex(A, 1) === 0x01
 @test getindex(A, 2) === missing
-@test getindex(A, 3) === missing
-@test getindex(A, 4) === missing
-@test getindex(A, 5) === missing
 @test getindex(A, 6) === 0x03
 @test getindex(A, 7) === missing
 @test getindex(A, 8) === missing