README.md in dydx-0.1.4 vs README.md in dydx-0.1.25

@@ -1,97 +1,160 @@
-# Dydx is new MATH Language on Ruby.
+# Dydx
+It always happens you want to differentiate some formulas with ruby. right?.....
 
-### Since you report a bug, I will fix it within 24 hours.
+After `inlcude Dydx` , ruby become like other language.
 
-The most important thing in this DSL is
+## Dydx is new math DSL in Ruby
 
-we can handle math in the same sense sense of the math on paper.
+I'm going to add for more explanation.
 
-ex. limit, trigonometric functions and logarithmic.
 
-## We should enjoy MATH even using PC.
-(to say nothing of using pen.)
 
-After `inlcude Dydx` , ruby become like other language.
-
-## Outline
 ```ruby:
 require 'dydx'
 include Dydx
 
-# Define the function. syntax is not good enough...
-f(x, y) <= x + x*y + y
+f(x) <= x ^ 2
 
-# simplify
-f(x, y) == x * (1 + y) + y
-=> true
+f(3)
+# => 9
 
-#part substitution
-f(a, 2) == 3*a + 2
-=> true
+f(x).to_s
+# => "( x ^ 2 )"
 
-f(1, a + b) == 1 + 2 * ( a + b )
-=> true
+f(x) == eval('f(x).to_s')
+# => true
 
+g(x) <= d/dx(f(x))
 
-# Differentiate
-d/dx(f(x, y)) == 1 + y
-=> true
+g(3)
+# => 6
 
-g(x) <= sin(x)
+# => '2 * x'
 
-d/dx(g(x)) == cos(x)
+S(f(x), dx)[0, 1]
+# => 0.3333333333333334
 
-# Integrate
-S(g(x), dx)[0, pi/2]
-=> 1.0
+( d/dx(log(x)) ).to_s
+# => "( 1 / x )"
+
+( d/dx(cos(x)) ).to_s
+# => "( - sin( x ) )"
+
+( d/dx(e ^ x) ).to_s
+# => "( e ^ x )"
+
+f(x) <= sin(x)
+S(f(x), dx)[0, Math::PI/2]
+# => 1.000000000021139
+
+f(x) <= (1.0 / ( ( 2.0 * pi ) ^ 0.5 ) ) * ( e ^ (- (x ^ 2) / 2) )
+S(f(x), dx)[-oo, oo]
+# => 0.9952054164466917
+
+f(x) <= x ^ 2
+
+f(a + b).to_s
+# => "( ( a + b ) ^ 2 )"
+
+g(a, b) <= f(a + b)
+
+g(a, b).to_s
+# => "( ( a + b ) ^ 2 )"
+
+g(2, 2)
+# => 16
+
+( d/da(g(a, b)) ).to_s
+=> "( 2 * ( a + b ) )"
 ```
 
 
-#### limit, trigonometric functions and logarithmic.
-```ruby:
 
-f(z) <= log(z)
-S(f(z), dz)[0,1]
-=> -Infinity
 
-d/dx(log(x)) == 1 / x
-=> true
 
-d/dx(cos(x)) == -cos(x)
-=> true
+```
+require 'dydx'
+include Dydx
 
-d/dx(e ** x) == e ** x
-=> true
+# There are three types of differential interface
 
-# standard normal distribution;
-f(x) <= (1.0 / ( 2.0 * pi ) ** 0.5) * e ** (- x ** 2 / 2)
-S(f(x), dx)[-oo, oo]
-=> 1.0
+( d/dx(x^2) ).to_s
+=> "( 2 * x )"
+
+log(z).d(z).to_s
+=> "( 1 / z )"
+
+$y = e ^ x
+(dy/dx).to_s
+=> "( e ^ x )"
+
 ```
 
-## Documents
-I'm going to write now...cominng soon....
+You may wonder why undefined `x` , `e` and `z` are handleable?
 
-### Module, class configuration
+`method_missing` solve this problem by converting undefine variable into internal class object.
 
+Like this.
+
 ```
-Dydx
-  |- Algebra
-  |      |- Operator
-  |      |   |- Interface
-  |      |   |- ....
-  |      |
-  |      |- Set
-  |      |- Formula
-  |      |- inverse
-  |
-  |- Function
-  |- Delta
-  |- Integrand
+ x + x
+=> #<Dydx::Algebra::Formula:0x007fb0a4039fb0 @f=#<Dydx::Algebra::Set::Num:0x007fb0a48169e0 @n=2>, @operator=:*, @g=:x>
+
+e
+=> #<Dydx::Algebra::Set::E:0x007fb0a383e9f0>
+
+log(sin(x))
+=> #<Dydx::Algebra::Set::Log:0x007fe7cd971528 @f=#<Dydx::Algebra::Set::Sin:0x007fe7cd971550 @x=:x>>
 ```
 
+And this DSL has strong simplify.
+
+```
+((x * y) + (z * x)).to_s
+=> "( x * ( y + z ) )"
+
+((x ^ y) / (x ^ z)).to_s
+=> "( x ^ ( y - z ) )"
+
+(x + x).to_s
+=> "( 2 * x )"
+```
+
+I show some differential calculus.
+
+```
+# pretermit '#to_s'
+
+d/dz(log(z))
+=> "( 1 / z )"
+
+d/dx(x^n)
+=> "( n * ( x ^ ( n - 1 ) ) )"
+
+$y = cos(x)
+dy/dx
+=> "( - sin( x ) )"
+
+$x = a * ( (t ^ 2) / 2 )
+dx/dt
+=> "( a * t )"
+
+d/dt(dx/dt)
+=>"a"
+
+((x ^ 2) * y).d(x)
+=> "( ( 2 * x ) * y )"
+
+((x ^ 2) * y).d(x).d(y)
+=> "( 2 * x )"
+
+```
+
+
+(That's wonderful!!!!! ..............)
+
 ## Installation
 
 Add this line to your application's Gemfile:
 
     gem 'dydx'
@@ -119,8 +182,8 @@
 ## Test
 
 run `bundle exec rake spec`
 
 ```
-Finished in 3.76 seconds
-325 examples, 0 failures
+Finished in 3.23 seconds
+309 examples, 0 failures
 ```