Add sqrt approximation for fractionals

@@ -62,6 +62,33 @@ impl Fractional {
     pub fn denominator(self) -> i64 {
     pub fn denominator(self) -> i64 {
         self.1
         self.1
     }
     }
+
+    pub fn sqrt(self) -> Self {
+        // find the sqrt of x in O(log x/2).
+        fn floor_sqrt(x :i64) -> i64 {
+            fn inner(l :i64, h :i64, x :i64) -> i64 {
+                if l > h {
+                    l - 1
+                } else {
+                    let m = (l + h) / 2;
+                    match x.cmp(&(m * m)) {
+                        Ordering::Equal   => m,
+                        Ordering::Less    => inner(l, m, x),
+                        Ordering::Greater => inner(m + 1, h, x),
+                    }
+                }
+            }
+
+            match x.cmp(&0) {
+                Ordering::Equal   => 0,
+                Ordering::Less    => -inner(1, -x / 2, -x),
+                Ordering::Greater => inner(1, x / 2, x),
+            }
+        }
+
+        let Fractional(n, d) = self;
+        Fractional(floor_sqrt(n), floor_sqrt(d)).reduce()
+    }
 }
 }
 
 
 impl fmt::Display for Fractional {
 impl fmt::Display for Fractional {

@@ -45,12 +45,19 @@ fn main() {
     let d :f64 = c.try_into().unwrap();
     let d :f64 = c.try_into().unwrap();
     let e :f64 = Fractional::try_into(c).unwrap();
     let e :f64 = Fractional::try_into(c).unwrap();
 
 
+    let f      = Fractional(9, 4);
+    let g      = Fractional(-9, 4);
+
     println!("         [i32] : {:?}"   , a);
     println!("         [i32] : {:?}"   , a);
     println!("  [Fractional] : {:?}"   , b);
     println!("  [Fractional] : {:?}"   , b);
     println!(" mean of [i32] : {}"     , c);
     println!(" mean of [i32] : {}"     , c);
     println!("        as f64 : {}"     , d);
     println!("        as f64 : {}"     , d);
     println!("    and as f64 : {}"     , e);
     println!("    and as f64 : {}"     , e);
     println!("  again as f64 : {}"     , TryInto::<f64>::try_into(c).unwrap());
     println!("  again as f64 : {}"     , TryInto::<f64>::try_into(c).unwrap());
+    println!("        sqrt f : {}"     , f.sqrt());
+    println!(" sqrt f as f64 : {}"     , TryInto::<f64>::try_into(f.sqrt()).unwrap());
+    println!("        sqrt g : {}"     , g.sqrt());
+    println!(" sqrt g as f64 : {}"     , TryInto::<f64>::try_into(g.sqrt()).unwrap());
     println!("        Rust π : {}"     , FPI);
     println!("        Rust π : {}"     , FPI);
     println!("             π : {} {}"  , TryInto::<f64>::try_into(PI).unwrap(), PI);
     println!("             π : {} {}"  , TryInto::<f64>::try_into(PI).unwrap(), PI);
     println!("    π as tuple : {:?}"   , TryInto::<(i32, i32)>::try_into(PI).unwrap());
     println!("    π as tuple : {:?}"   , TryInto::<(i32, i32)>::try_into(PI).unwrap());