diff --git a/libio/iofdopen.c b/libio/iofdopen.c
index 5167220aa2..c86cf0f3ca 100644
--- a/libio/iofdopen.c
+++ b/libio/iofdopen.c
@@ -124,8 +124,19 @@ _IO_new_fdopen (fd, mode)
 #ifdef _IO_MTSAFE_IO
   new_f->fp.file._lock = &new_f->lock;
 #endif
-  _IO_no_init (&new_f->fp.file, 0, 0, &new_f->wd, &INTUSE(_IO_wfile_jumps));
-  _IO_JUMPS (&new_f->fp) = &INTUSE(_IO_file_jumps);
+  /* Set up initially to use the `maybe_mmap' jump tables rather than using
+     __fopen_maybe_mmap to do it, because we need them in place before we
+     call _IO_file_attach or else it will allocate a buffer immediately.  */
+  _IO_no_init (&new_f->fp.file, 0, 0, &new_f->wd,
+#ifdef _G_HAVE_MMAP
+	       (read_write & _IO_NO_WRITES) ? &_IO_wfile_jumps_maybe_mmap :
+#endif
+	       &INTUSE(_IO_wfile_jumps));
+  _IO_JUMPS (&new_f->fp) =
+#ifdef _G_HAVE_MMAP
+    (read_write & _IO_NO_WRITES) ? &_IO_file_jumps_maybe_mmap :
+#endif
+      &INTUSE(_IO_file_jumps);
   INTUSE(_IO_file_init) (&new_f->fp);
 #if  !_IO_UNIFIED_JUMPTABLES
   new_f->fp.vtable = NULL;
@@ -143,7 +154,7 @@ _IO_new_fdopen (fd, mode)
     _IO_mask_flags (&new_f->fp.file, read_write,
 		    _IO_NO_READS+_IO_NO_WRITES+_IO_IS_APPENDING);
 
-  return __fopen_maybe_mmap (&new_f->fp.file);
+  return &new_f->fp.file;
 }
 INTDEF2(_IO_new_fdopen, _IO_fdopen)
 
diff --git a/libio/iofopen.c b/libio/iofopen.c
index c1681a4a10..00a2c5f073 100644
--- a/libio/iofopen.c
+++ b/libio/iofopen.c
@@ -43,50 +43,16 @@ __fopen_maybe_mmap (fp)
 #ifdef _G_HAVE_MMAP
   if (fp->_flags & _IO_NO_WRITES)
     {
-      /* We use the file in read-only mode.  This could mean we can
-	 mmap the file and use it without any copying.  But not all
-	 file descriptors are for mmap-able objects and on 32-bit
-	 machines we don't want to map files which are too large since
-	 this would require too much virtual memory.  */
-      struct _G_stat64 st;
+      /* Since this is read-only, we might be able to mmap the contents
+	 directly.  We delay the decision until the first read attempt by
+	 giving it a jump table containing functions that choose mmap or
+	 vanilla file operations and reset the jump table accordingly.  */
 
-      if (_IO_SYSSTAT (fp, &st) == 0
-	  && S_ISREG (st.st_mode) && st.st_size != 0
-	  /* Limit the file size to 1MB for 32-bit machines.  */
-	  && (sizeof (ptrdiff_t) > 4 || st.st_size < 1*1024*1024)
-	  /* Sanity check.  */
-	  && (fp->_offset == _IO_pos_BAD || fp->_offset <= st.st_size))
-	{
-	  /* Try to map the file.  */
-	  void *p;
-
-# ifdef _G_MMAP64
-	  p = _G_MMAP64 (NULL, st.st_size, PROT_READ, MAP_PRIVATE,
-			 fp->_fileno, 0);
-# else
-	  p = __mmap (NULL, st.st_size, PROT_READ, MAP_PRIVATE,
-		      fp->_fileno, 0);
-# endif
-	  if (p != MAP_FAILED)
-	    {
-	      /* OK, we managed to map the file.  Set the buffer up
-		 and use a special jump table with simplified
-		 underflow functions which never tries to read
-		 anything from the file.  */
-	      INTUSE(_IO_setb) (fp, p, (char *) p + st.st_size, 0);
-
-	      if (fp->_offset == _IO_pos_BAD)
-		fp->_offset = 0;
-
-	      _IO_setg (fp,  p, p + fp->_offset, p + fp->_offset);
-
-	      if (fp->_mode <= 0)
-		_IO_JUMPS ((struct _IO_FILE_plus *) fp) = &_IO_file_jumps_mmap;
-	      else
-		_IO_JUMPS ((struct _IO_FILE_plus *) fp) = &_IO_wfile_jumps_mmap;
-	      fp->_wide_data->_wide_vtable = &_IO_wfile_jumps_mmap;
-	    }
-	}
+      if (fp->_mode <= 0)
+	_IO_JUMPS ((struct _IO_FILE_plus *) fp) = &_IO_file_jumps_maybe_mmap;
+      else
+	_IO_JUMPS ((struct _IO_FILE_plus *) fp) = &_IO_wfile_jumps_maybe_mmap;
+      fp->_wide_data->_wide_vtable = &_IO_wfile_jumps_maybe_mmap;
     }
 #endif
   return fp;
diff --git a/libio/libioP.h b/libio/libioP.h
index bfb36c30ef..cb9e918468 100644
--- a/libio/libioP.h
+++ b/libio/libioP.h
@@ -425,8 +425,10 @@ extern void _IO_default_imbue __P ((_IO_FILE *, void *));
 
 extern struct _IO_jump_t _IO_file_jumps;
 extern struct _IO_jump_t _IO_file_jumps_mmap attribute_hidden;
+extern struct _IO_jump_t _IO_file_jumps_maybe_mmap attribute_hidden;
 extern struct _IO_jump_t _IO_wfile_jumps attribute_hidden;
 extern struct _IO_jump_t _IO_wfile_jumps_mmap attribute_hidden;
+extern struct _IO_jump_t _IO_wfile_jumps_maybe_mmap attribute_hidden;
 extern struct _IO_jump_t _IO_old_file_jumps attribute_hidden;
 extern struct _IO_jump_t _IO_streambuf_jumps;
 extern struct _IO_jump_t _IO_proc_jumps attribute_hidden;
@@ -498,6 +500,8 @@ extern _IO_FILE* _IO_file_setbuf __P ((_IO_FILE *, char *, _IO_ssize_t));
 extern _IO_off64_t _IO_file_seekoff __P ((_IO_FILE *, _IO_off64_t, int, int));
 extern _IO_off64_t _IO_file_seekoff_mmap __P ((_IO_FILE *, _IO_off64_t, int,
 					       int));
+extern _IO_off64_t _IO_file_seekoff_maybe_mmap __P ((_IO_FILE *, _IO_off64_t,
+						     int, int));
 extern _IO_size_t _IO_file_xsputn __P ((_IO_FILE *, const void *, _IO_size_t));
 extern _IO_size_t _IO_file_xsgetn __P ((_IO_FILE *, void *, _IO_size_t));
 extern int _IO_file_stat __P ((_IO_FILE *, void *));
@@ -505,6 +509,7 @@ extern int _IO_file_close __P ((_IO_FILE *));
 extern int _IO_file_close_mmap __P ((_IO_FILE *));
 extern int _IO_file_underflow __P ((_IO_FILE *));
 extern int _IO_file_underflow_mmap __P ((_IO_FILE *));
+extern int _IO_file_underflow_maybe_mmap __P ((_IO_FILE *));
 extern int _IO_file_overflow __P ((_IO_FILE *, int));
 #define _IO_file_is_open(__fp) ((__fp)->_fileno != -1)
 extern void _IO_file_init __P ((struct _IO_FILE_plus *));