Problem:
Score : 600 points

Problem Statement

Given are integers L and R. Find the number, modulo 10^9 + 7, of pairs of integers (x, y) (L \leq x \leq y \leq R) such that the remainder when y is divided by x is equal to y \mbox{ XOR } x.

What is \mbox{ XOR }?

The XOR of integers A and B, A \mbox{ XOR } B, is defined as follows:

When A \mbox{ XOR } B is written in base two, the digit in the 2^k's place (k \geq 0) is 1 if either A or B, but not both, has 1 in the 2^k's place, and 0 otherwise.

For example, 3 \mbox{ XOR } 5 = 6. (In base two: 011 \mbox{ XOR } 101 = 110.)

Constraints

1 \leq L \leq R \leq 10^{18}

Input

Input is given from Standard Input in the following format:

L R

Output

Print the number of pairs of integers (x, y) (L \leq x \leq y \leq R) satisfying the condition, modulo 10^9 + 7.

Sample Input 1

2 3

Sample Output 1

3

Three pairs satisfy the condition: (2, 2), (2, 3), and (3, 3).

Sample Input 2

10 100

Sample Output 2

604

Sample Input 3

1 1000000000000000000

Sample Output 3

68038601

Be sure to compute the number modulo 10^9 + 7.

Solution in Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           ;; enclose the form with VALUES to avoid being captured by LOOP macro
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Arithmetic operations with static modulus
;;;

(defmacro define-mod-operations (divisor)
  `(progn
     (defun mod* (&rest args)
       (reduce (lambda (x y) (mod (* x y) ,divisor)) args))

     (sb-c:define-source-transform mod* (&rest args)
       (if (null args)
           1
           (reduce (lambda (x y) `(mod (* ,x ,y) ,',divisor)) args)))

     (defun mod+ (&rest args)
       (reduce (lambda (x y) (mod (+ x y) ,divisor)) args))

     (sb-c:define-source-transform mod+ (&rest args)
       (if (null args)
           0
           (reduce (lambda (x y) `(mod (+ ,x ,y) ,',divisor)) args)))

     (define-modify-macro incfmod (delta)
       (lambda (x y) (mod (+ x y) ,divisor)))

     (define-modify-macro decfmod (delta)
       (lambda (x y) (mod (- x y) ,divisor)))

     (define-modify-macro mulfmod (multiplier)
       (lambda (x y) (mod (* x y) ,divisor)))))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(define-mod-operations +mod+)
(defun main ()
  (let* ((l (read))
         (r (read))
         (dp (make-array '(62 2 2 2) :element-type 'uint32 :initial-element 0)))
    (setf (aref dp 61 0 0 0) 1)
    (loop
      for i from 61 above 0
      for li = (ldb (byte 1 (- i 1)) l)
      for ri = (ldb (byte 1 (- i 1)) r)
      do (dotimes (x 2)
           (dotimes (y 2)
             (dotimes (z 2)
               (macrolet ((inc (new-x new-y new-z)
                            `(incfmod (aref dp (- i 1) ,new-x ,new-y ,new-z)
                                      (aref dp i x y z))))
                 (cond ((and (= x 0) (= y 0) (= z 0))
                        (cond ((and (= li 0) (= ri 0))
                               (inc 0 0 0))
                              ((and (= li 0) (= ri 1))
                               (inc 0 1 0)
                               (inc 1 0 1))
                              ((and (= li 1) (= ri 0)))
                              ((and (= li 1) (= ri 1))
                               (inc 0 0 1))))
                       ((and (= x 0) (= y 0) (= z 1))
                        (cond ((and (= li 0) (= ri 0))
                               (inc 0 0 1))
                              ((and (= li 0) (= ri 1))
                               (inc 0 1 1)
                               (inc 0 0 1)
                               (inc 1 0 1))
                              ((and (= li 1) (= ri 0)))
                              ((and (= li 1) (= ri 1))
                               (inc 0 0 1))))
                       ((and (= x 0) (= y 1) (= z 0))
                        (cond ((and (= li 0) (= ri 0))
                               (inc 0 1 0)
                               (inc 1 1 1))
                              ((and (= li 0) (= ri 1))
                               (inc 0 1 0)
                               (inc 1 1 1))
                              ((and (= li 1) (= ri 0))
                               (inc 0 1 1))
                              ((and (= li 1) (= ri 1))
                               (inc 0 1 1))))
                       ((and (= x 0) (= y 1) (= z 1))
                        (cond ((and (= li 0) (= ri 0))
                               (inc 0 1 1)
                               (inc 0 1 1)
                               (inc 1 1 1))
                              ((and (= li 0) (= ri 1))
                               (inc 0 1 1)
                               (inc 0 1 1)
                               (inc 1 1 1))
                              ((and (= li 1) (= ri 0))
                               (inc 0 1 1))
                              ((and (= li 1) (= ri 1))
                               (inc 0 1 1))))
                       ((and (= x 1) (= y 0) (= z 0))
                        (cond ((and (= li 0) (= ri 0))
                               (inc 1 0 0))
                              ((and (= li 0) (= ri 1))
                               (inc 1 1 0)
                               (inc 1 0 1))
                              ((and (= li 1) (= ri 0))
                               (inc 1 0 0))
                              ((and (= li 1) (= ri 1))
                               (inc 1 1 0)
                               (inc 1 0 1))))
                       ((and (= x 1) (= y 0) (= z 1))
                        (cond ((and (= li 0) (= ri 0))
                               (inc 1 0 1))
                              ((and (= li 0) (= ri 1))
                               (inc 1 1 1)
                               (inc 1 0 1)
                               (inc 1 0 1))
                              ((and (= li 1) (= ri 0))
                               (inc 1 0 1))
                              ((and (= li 1) (= ri 1))
                               (inc 1 1 1)
                               (inc 1 0 1)
                               (inc 1 0 1))))
                       ((and (= x 1) (= y 1) (= z 0))
                        (inc 1 1 0)
                        (inc 1 1 1))
                       ((and (= x 1) (= y 1) (= z 1))
                        (inc 1 1 1)
                        (inc 1 1 1)
                        (inc 1 1 1))))))))
    (dbg dp)
    (println
     (mod+ (aref dp 0 0 0 0)
           (aref dp 0 0 0 1)
           (aref dp 0 0 1 0)
           (aref dp 0 0 1 1)
           (aref dp 0 1 0 0)
           (aref dp 0 1 0 1)
           (aref dp 0 1 1 0)
           (aref dp 0 1 1 1)))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    (run-program "C:/msys64/usr/bin/cat.exe" '("/dev/clipboard") :output out)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "2 3
"
    "3
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "10 100
"
    "604
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "1 1000000000000000000
"
    "68038601
")))