diff --git a/Modules/_io/fileio.c b/Modules/_io/fileio.c
index b1d492b..be5c9f8 100644
--- a/Modules/_io/fileio.c
+++ b/Modules/_io/fileio.c
@@ -43,12 +43,6 @@
 #define SMALLCHUNK BUFSIZ
 #endif
 
-#if SIZEOF_INT < 4
-#define BIGCHUNK  (512 * 32)
-#else
-#define BIGCHUNK  (512 * 1024)
-#endif
-
 typedef struct {
     PyObject_HEAD
     int fd;
@@ -565,15 +559,10 @@
         }
     }
 #endif
-    if (currentsize > SMALLCHUNK) {
-        /* Keep doubling until we reach BIGCHUNK;
-           then keep adding BIGCHUNK. */
-        if (currentsize <= BIGCHUNK)
-            return currentsize + currentsize;
-        else
-            return currentsize + BIGCHUNK;
-    }
-    return currentsize + SMALLCHUNK;
+    /* Expand the buffer by an amount proportional to the current size,
+       giving us amortized linear-time behavior. Use a less-than-double
+       growth factor to avoid excessive allocation. */
+    return currentsize + (currentsize >> 3) + 6;
 }
 
 static PyObject *
diff --git a/Modules/bz2module.c b/Modules/bz2module.c
index 3e55202..a671e8d 100644
--- a/Modules/bz2module.c
+++ b/Modules/bz2module.c
@@ -218,25 +218,14 @@
 #define SMALLCHUNK BUFSIZ
 #endif
 
-#if SIZEOF_INT < 4
-#define BIGCHUNK  (512 * 32)
-#else
-#define BIGCHUNK  (512 * 1024)
-#endif
-
 /* This is a hacked version of Python's fileobject.c:new_buffersize(). */
 static size_t
 Util_NewBufferSize(size_t currentsize)
 {
-    if (currentsize > SMALLCHUNK) {
-        /* Keep doubling until we reach BIGCHUNK;
-           then keep adding BIGCHUNK. */
-        if (currentsize <= BIGCHUNK)
-            return currentsize + currentsize;
-        else
-            return currentsize + BIGCHUNK;
-    }
-    return currentsize + SMALLCHUNK;
+    /* Expand the buffer by an amount proportional to the current size,
+       giving us amortized linear-time behavior. Use a less-than-double
+       growth factor to avoid excessive allocation. */
+    return currentsize + (currentsize >> 3) + 6;
 }
 
 /* This is a hacked version of Python's fileobject.c:get_line(). */