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 Rust (1.15.1) (judge verdict: Accepted)

pub trait Readable {
    type Output;
    fn words_count() -> usize;
    fn read_words(words: &[&str]) -> Result<Self::Output, String>;
}
#[macro_export]
macro_rules! readable {
    ( $ t : ty , $ words_count : expr , |$ words : ident | $ read_words : expr ) => {
        impl Readable for $t {
            type Output = $t;
            fn words_count() -> usize {
                $words_count
            }
            fn read_words($words: &[&str]) -> Result<$t, String> {
                Ok($read_words)
            }
        }
    };
}
readable!((), 1, |_ss| ());
readable!(String, 1, |ss| ss[0].to_string());
impl Readable for char {
    type Output = char;
    fn words_count() -> usize {
        1
    }
    fn read_words(words: &[&str]) -> Result<char, String> {
        let chars: Vec<char> = words[0].chars().collect();
        if chars.len() == 1 {
            Ok(chars[0])
        } else {
            Err(format!("cannot parse `{}` as a char", words[0]))
        }
    }
}
pub struct Chars();
impl Readable for Chars {
    type Output = Vec<char>;
    fn words_count() -> usize {
        1
    }
    fn read_words(words: &[&str]) -> Result<Vec<char>, String> {
        Ok(words[0].chars().collect())
    }
}
macro_rules ! impl_readable_for_ints { ( $ ( $ t : ty ) * ) => { $ ( impl Readable for $ t { type Output = Self ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result <$ t , String > { use std :: str :: FromStr ; <$ t >:: from_str ( words [ 0 ] ) . map_err ( | _ | { format ! ( "cannot parse `{}` as {}" , words [ 0 ] , stringify ! ( $ t ) ) } ) } } ) * } ; }
impl_readable_for_ints ! ( i8 u8 i16 u16 i32 u32 i64 u64 isize usize f32 f64 ) ;
macro_rules ! define_one_origin_int_types { ( $ new_t : ident $ int_t : ty ) => { # [ doc = " Converts 1-origin integer into 0-origin when read from stdin." ] # [ doc = "" ] # [ doc = " # Example" ] # [ doc = "" ] # [ doc = " ```no_run" ] # [ doc = " # #[macro_use] extern crate atcoder_snippets;" ] # [ doc = " # use atcoder_snippets::read::*;" ] # [ doc = " // Stdin: \"1\"" ] # [ doc = " read!(a = usize_);" ] # [ doc = " assert_eq!(a, 0);" ] # [ doc = " ```" ] # [ allow ( non_camel_case_types ) ] pub struct $ new_t ; impl Readable for $ new_t { type Output = $ int_t ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { <$ int_t >:: read_words ( words ) . map ( | n | n - 1 ) } } } ; ( $ new_t : ident $ int_t : ty ; $ ( $ inner_new_t : ident $ inner_int_t : ty ) ;* ) => { define_one_origin_int_types ! ( $ new_t $ int_t ) ; define_one_origin_int_types ! ( $ ( $ inner_new_t $ inner_int_t ) ;* ) ; } ; }
define_one_origin_int_types ! ( u8_ u8 ; u16_ u16 ; u32_ u32 ; u64_ u64 ; usize_ usize ) ;
macro_rules ! impl_readable_for_tuples { ( $ t : ident $ var : ident ) => ( ) ; ( $ t : ident $ var : ident ; $ ( $ inner_t : ident $ inner_var : ident ) ;* ) => { impl_readable_for_tuples ! ( $ ( $ inner_t $ inner_var ) ;* ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,*> Readable for ( $ t , $ ( $ inner_t ) ,* ) { type Output = ( <$ t >:: Output , $ ( <$ inner_t >:: Output ) ,* ) ; fn words_count ( ) -> usize { let mut n = <$ t >:: words_count ( ) ; $ ( n += <$ inner_t >:: words_count ( ) ; ) * n } # [ allow ( unused_assignments ) ] fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { let mut start = 0 ; let $ var = <$ t >:: read_words ( & words [ start .. start +<$ t >:: words_count ( ) ] ) ?; start += <$ t >:: words_count ( ) ; $ ( let $ inner_var = <$ inner_t >:: read_words ( & words [ start .. start +<$ inner_t >:: words_count ( ) ] ) ?; start += <$ inner_t >:: words_count ( ) ; ) * Ok ( ( $ var , $ ( $ inner_var ) ,* ) ) } } } ; }
impl_readable_for_tuples ! ( T8 x8 ; T7 x7 ; T6 x6 ; T5 x5 ; T4 x4 ; T3 x3 ; T2 x2 ; T1 x1 ) ;
pub trait ReadableFromLine {
    type Output;
    fn read_line(line: &str) -> Result<Self::Output, String>;
}
fn split_into_words(line: &str) -> Vec<&str> {
    #[allow(deprecated)]
    line.trim_right_matches('\n').split_whitespace().collect()
}
impl<T: Readable> ReadableFromLine for T {
    type Output = T::Output;
    fn read_line(line: &str) -> Result<T::Output, String> {
        let words = split_into_words(line);
        if words.len() != T::words_count() {
            return Err(format!(
                "line `{}` has {} words, expected {}",
                line,
                words.len(),
                T::words_count()
            ));
        }
        T::read_words(&words)
    }
}
macro_rules ! impl_readable_from_line_for_tuples_with_from_iterator { ( $ u : ident : $ ( + $ bound : path ) * => $ seq_in : ty , $ seq_out : ty ; $ t : ident $ var : ident ) => { impl <$ u : Readable > ReadableFromLine for $ seq_in where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = $ seq_out ; fn read_line ( line : & str ) -> Result <$ seq_out , String > { let n = $ u :: words_count ( ) ; let words = split_into_words ( line ) ; if words . len ( ) % n != 0 { return Err ( format ! ( "line `{}` has {} words, expected multiple of {}" , line , words . len ( ) , n ) ) ; } let mut result = Vec :: new ( ) ; for chunk in words . chunks ( n ) { match $ u :: read_words ( chunk ) { Ok ( v ) => result . push ( v ) , Err ( msg ) => { let flagment_msg = if n == 1 { format ! ( "word {}" , result . len ( ) ) } else { let l = result . len ( ) ; format ! ( "words {}-{}" , n * l + 1 , ( n + 1 ) * l ) } ; return Err ( format ! ( "{} of line `{}`: {}" , flagment_msg , line , msg ) ) ; } } } Ok ( result . into_iter ( ) . collect ( ) ) } } impl < T : Readable , $ u : Readable > ReadableFromLine for ( T , $ seq_in ) where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = ( T :: Output , $ seq_out ) ; fn read_line ( line : & str ) -> Result < Self :: Output , String > { let n = T :: words_count ( ) ; # [ allow ( deprecated ) ] let trimmed = line . trim_right_matches ( '\n' ) ; let words_and_rest : Vec <& str > = trimmed . splitn ( n + 1 , ' ' ) . collect ( ) ; if words_and_rest . len ( ) < n { return Err ( format ! ( "line `{}` has {} words, expected at least {}" , line , words_and_rest . len ( ) , n ) ) ; } let words = & words_and_rest [ .. n ] ; let empty_str = "" ; let rest = words_and_rest . get ( n ) . unwrap_or ( & empty_str ) ; Ok ( ( T :: read_words ( words ) ?, <$ seq_in >:: read_line ( rest ) ? ) ) } } } ; ( $ u : ident : $ ( + $ bound : path ) * => $ seq_in : ty , $ seq_out : ty ; $ t : ident $ var : ident , $ ( $ inner_t : ident $ inner_var : ident ) ,+ ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : $ ( + $ bound ) * => $ seq_in , $ seq_out ; $ ( $ inner_t $ inner_var ) ,+ ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,+ , $ u : Readable > ReadableFromLine for ( $ t , $ ( $ inner_t ) ,+ , $ seq_in ) where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = ( $ t :: Output , $ ( $ inner_t :: Output ) ,+ , $ seq_out ) ; fn read_line ( line : & str ) -> Result < Self :: Output , String > { let mut n = $ t :: words_count ( ) ; $ ( n += $ inner_t :: words_count ( ) ; ) + # [ allow ( deprecated ) ] let trimmed = line . trim_right_matches ( '\n' ) ; let words_and_rest : Vec <& str > = trimmed . splitn ( n + 1 , ' ' ) . collect ( ) ; if words_and_rest . len ( ) < n { return Err ( format ! ( "line `{}` has {} words, expected at least {}" , line , words_and_rest . len ( ) , n ) ) ; } let words = & words_and_rest [ .. n ] ; let empty_str = "" ; let rest = words_and_rest . get ( n ) . unwrap_or ( & empty_str ) ; let ( $ var , $ ( $ inner_var ) ,* ) = < ( $ t , $ ( $ inner_t ) ,+ ) >:: read_words ( words ) ?; Ok ( ( $ var , $ ( $ inner_var ) ,* , <$ seq_in >:: read_line ( rest ) ? ) ) } } } ; }
#[macro_export]
macro_rules ! readable_collection { ( $ u : ident => $ collection_in : ty , $ collection_out : ty ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : => $ collection_in , $ collection_out ; T8 x8 , T7 x7 , T6 x6 , T5 x5 , T4 t4 , T3 t3 , T2 t2 , T1 t1 ) ; } ; ( $ u : ident : $ ( $ bound : path ) ,* => $ collection_in : ty , $ collection_out : ty ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : $ ( + $ bound ) * => $ collection_in , $ collection_out ; T8 x8 , T7 x7 , T6 x6 , T5 x5 , T4 t4 , T3 t3 , T2 t2 , T1 t1 ) ; } }
readable_collection ! ( U => Vec < U >, Vec < U :: Output > ) ;
readable_collection ! ( U : Eq , std :: hash :: Hash => std :: collections :: HashSet < U >, std :: collections :: HashSet < U :: Output > ) ;
pub fn read<T: ReadableFromLine>() -> T::Output {
    let mut line = String::new();
    std::io::stdin().read_line(&mut line).unwrap();
    T::read_line(&line).unwrap()
}
#[macro_export]
macro_rules ! read { ( ) => { let mut line = String :: new ( ) ; std :: io :: stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; } ; ( $ pat : pat = $ t : ty ) => { let $ pat = read ::<$ t > ( ) ; } ; ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { read ! ( ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) ) ; } ; }
#[macro_export]
macro_rules ! readls { ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { $ ( read ! ( $ pat = $ t ) ; ) * } ; }
pub fn readx<T: ReadableFromLine>() -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result = stdin
        .lock()
        .lines()
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    result
}
#[macro_export]
macro_rules ! readx_loop { ( |$ pat : pat = $ t : ty | $ body : expr ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; for line in stdin . lock ( ) . lines ( ) { let line = line . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } ; ( |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readx_loop ! ( | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub fn readn<T: ReadableFromLine>(n: usize) -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result: Vec<T::Output> = stdin
        .lock()
        .lines()
        .take(n)
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    if result.len() < n {
        panic!(
            "expected reading {} lines, but only {} lines are read",
            n,
            result.len()
        );
    }
    result
}
#[macro_export]
macro_rules ! readn_loop { ( $ n : expr , |$ pat : pat = $ t : ty | $ body : expr ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; { let mut lock = stdin . lock ( ) ; for _ in 0 ..$ n { let mut line = String :: new ( ) ; lock . read_line ( & mut line ) . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } } ; ( $ n : expr , |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readn_loop ! ( $ n , | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub trait Words {
    fn read<T: Readable>(&self) -> T::Output;
}
impl<'a> Words for [&'a str] {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(self).unwrap()
    }
}
impl<'a> Words for &'a str {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(&[self]).unwrap()
    }
}

pub type ModPBase = u64;
pub const MODULUS: ModPBase = 1_000_000_007;
#[derive(Clone, Copy, PartialEq, Eq)]
pub struct ModP {
    base: ModPBase,
}
impl ModP {
    pub fn new(n: ModPBase) -> ModP {
        if !cfg!(test) {
            assert!(
                MODULUS != 7,
                "Set const MODULUS to the value provided by the problem."
            );
        }
        ModP { base: n % MODULUS }
    }
    pub unsafe fn new_unchecked(n: ModPBase) -> ModP {
        ModP { base: n }
    }
    pub fn base(&self) -> ModPBase {
        self.base
    }
    pub fn pow(self, exp: ModPBase) -> ModP {
        if exp == 0 {
            ModP::new(1)
        } else {
            let sub = self.pow(exp / 2);
            if exp % 2 == 0 {
                sub * sub
            } else {
                self * sub * sub
            }
        }
    }
    pub fn inv(self) -> ModP {
        assert!(self.base() != 0);
        self.pow(MODULUS - 2)
    }
}
pub fn modp(x: u64) -> ModP {
    ModP::new(x)
}
impl std::fmt::Display for ModP {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.base())
    }
}
impl std::fmt::Debug for ModP {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{} mod P", self.base())
    }
}
impl PartialEq<ModPBase> for ModP {
    fn eq(&self, other: &ModPBase) -> bool {
        self.base() == other % MODULUS
    }
}
impl PartialEq<ModP> for ModPBase {
    fn eq(&self, other: &ModP) -> bool {
        self % MODULUS == other.base() % MODULUS
    }
}
impl std::ops::Add for ModP {
    type Output = ModP;
    fn add(self, rhs: ModP) -> ModP {
        ModP {
            base: (self.base() + rhs.base() % MODULUS) % MODULUS,
        }
    }
}
impl std::ops::Add<ModPBase> for ModP {
    type Output = ModP;
    fn add(self, rhs: ModPBase) -> ModP {
        self + ModP::new(rhs)
    }
}
impl std::ops::Add<ModP> for ModPBase {
    type Output = ModP;
    fn add(self, rhs: ModP) -> ModP {
        ModP::new(self) + rhs.base()
    }
}
impl std::ops::AddAssign for ModP {
    fn add_assign(&mut self, rhs: ModP) {
        *self = *self + rhs
    }
}
impl std::ops::AddAssign<ModPBase> for ModP {
    fn add_assign(&mut self, rhs: ModPBase) {
        *self = *self + ModP::new(rhs)
    }
}
impl std::ops::Neg for ModP {
    type Output = ModP;
    fn neg(self) -> ModP {
        ModP::new(MODULUS - self.base())
    }
}
impl std::ops::Sub for ModP {
    type Output = ModP;
    fn sub(self, rhs: ModP) -> ModP {
        self + (-rhs)
    }
}
impl std::ops::Sub<ModPBase> for ModP {
    type Output = ModP;
    fn sub(self, rhs: ModPBase) -> ModP {
        self - ModP::new(rhs)
    }
}
impl std::ops::Sub<ModP> for ModPBase {
    type Output = ModP;
    fn sub(self, rhs: ModP) -> ModP {
        ModP::new(self) - rhs
    }
}
impl std::ops::SubAssign for ModP {
    fn sub_assign(&mut self, rhs: ModP) {
        *self = *self - rhs;
    }
}
impl std::ops::SubAssign<ModPBase> for ModP {
    fn sub_assign(&mut self, rhs: ModPBase) {
        *self = *self - ModP::new(rhs)
    }
}
impl std::ops::Mul for ModP {
    type Output = ModP;
    fn mul(self, rhs: ModP) -> ModP {
        ModP {
            base: self.base() * (rhs.base() % MODULUS) % MODULUS,
        }
    }
}
impl std::ops::Mul<ModPBase> for ModP {
    type Output = ModP;
    fn mul(self, rhs: ModPBase) -> ModP {
        self * ModP::new(rhs)
    }
}
impl std::ops::Mul<ModP> for ModPBase {
    type Output = ModP;
    fn mul(self, rhs: ModP) -> ModP {
        ModP::new(self) * rhs.base()
    }
}
impl std::ops::MulAssign for ModP {
    fn mul_assign(&mut self, rhs: ModP) {
        *self = *self * rhs
    }
}
impl std::ops::MulAssign<ModPBase> for ModP {
    fn mul_assign(&mut self, rhs: ModPBase) {
        *self = *self * ModP::new(rhs)
    }
}
impl std::ops::Div for ModP {
    type Output = ModP;
    fn div(self, rhs: ModP) -> ModP {
        self * rhs.inv()
    }
}
impl std::ops::Div<ModPBase> for ModP {
    type Output = ModP;
    fn div(self, rhs: ModPBase) -> ModP {
        self * ModP::new(rhs).inv()
    }
}
impl std::ops::Div<ModP> for ModPBase {
    type Output = ModP;
    fn div(self, rhs: ModP) -> ModP {
        ModP::new(self) * rhs.inv()
    }
}
impl std::ops::DivAssign for ModP {
    fn div_assign(&mut self, rhs: ModP) {
        *self = *self / rhs;
    }
}
impl std::ops::DivAssign<ModPBase> for ModP {
    fn div_assign(&mut self, rhs: ModPBase) {
        *self = *self / ModP::new(rhs)
    }
}
impl std::iter::Sum for ModP {
    fn sum<I: Iterator<Item = ModP>>(iter: I) -> ModP {
        let mut ans = 0;
        for n in iter {
            ans += n.base();
        }
        ModP::new(ans)
    }
}
impl<'a> std::iter::Sum<&'a ModP> for ModP {
    fn sum<I: Iterator<Item = &'a ModP>>(iter: I) -> ModP {
        let mut ans = 0;
        for n in iter {
            ans += n.base();
        }
        ModP::new(ans)
    }
}
impl std::iter::Product for ModP {
    fn product<I: Iterator<Item = ModP>>(iter: I) -> ModP {
        let mut ans = unsafe { ModP::new_unchecked(1) };
        for n in iter {
            ans *= n;
        }
        ans
    }
}
impl<'a> std::iter::Product<&'a ModP> for ModP {
    fn product<I: Iterator<Item = &'a ModP>>(iter: I) -> ModP {
        let mut ans = unsafe { ModP::new_unchecked(1) };
        for &n in iter {
            ans *= n;
        }
        ans
    }
}
readable!(ModP, 1, |ws| ModP::new(ws[0].read::<u64>()));

#[macro_export]
#[cfg(local)]
macro_rules! dbg {
    ( ) => {{
        use std::io::{self, Write};
        writeln!(io::stderr(), "{}: dbg", line!()).unwrap();
    }};
    ( $ e : expr ) => {{
        use std::io::{self, Write};
        let result = $e;
        writeln!(
            io::stderr(),
            "{}: {} = {:?}",
            line!(),
            stringify!($e),
            result
        )
            .unwrap();
        result
    }};
}
#[macro_export]
#[cfg(not(local))]
macro_rules! dbg {
    ( ) => {};
    ( $ e : expr ) => {
        $e
    };
}

fn to_bin(mut x: u64) -> Vec<u8> {
    let mut res = Vec::new();
    while x != 0 {
        res.push((x%2) as u8);
        x /= 2;
    }
    res.reverse();
    res
}

fn count(
    l: &[u8], r: &[u8], xt: bool, yt: bool, valid: bool,
    memo: &mut Vec<Vec<Option<ModP>>>
) -> ModP {
    let flag_index = (xt as usize) * 4 + (yt as usize) * 2 + (valid as usize);
    memo[flag_index][l.len()].unwrap_or_else(|| {
        let ans = if l.is_empty() { modp(1) } else {
            let (&l0, l_tl) = l.split_first().unwrap();
            let (&r0, r_tl) = r.split_first().unwrap();
            if yt && r0 == 0 {
                if xt && l0 == 1 {
                    modp(0)
                } else {
                    count(l_tl, r_tl, xt, yt, valid, memo) // (x, y) = (0, 0)
                }
            } else {
                if xt && l0 == 1 {
                    count(l_tl, r_tl, xt, yt, true, memo) // (x, y) = (1, 1)
                } else {
                    count(l_tl, r_tl, xt, false, valid, memo) // (x, y) = (0, 0)
                        + if valid { count(l_tl, r_tl, xt, yt, true, memo) } else { modp(0) } // (x, y) = (0, 1)
                    + count(l_tl, r_tl, false, yt, true, memo) // (x, y) = (1, 1)
                }
            }
        };
        memo[flag_index][l.len()] = Some(ans);
        ans
    })
}

fn main() {
    read!(l = u64, r = u64);
    let r_bin = to_bin(r);
    let l_bin_suff = to_bin(l);
    let mut l_bin = vec![0; r_bin.len() - l_bin_suff.len()];
    l_bin.extend_from_slice(&l_bin_suff);
    let mut memo = vec![vec![None; l_bin.len() + 1]; 8];
    println!("{}", count(&l_bin, &r_bin, true, true, false, &mut memo));
}