diff --git a/stl/mat.nas b/stl/mat.nas
index 600fcf9..0fcc84b 100644
--- a/stl/mat.nas
+++ b/stl/mat.nas
@@ -13,10 +13,9 @@ var mat=func(width,height) {
 }
 
 var rand_init=func(a) {
-    srand();
     var ref=a.mat;
     forindex(var i;ref) {
-        ref[i]=rand();
+        ref[i]=rand()*2-1;
     }
 }
 
@@ -64,6 +63,23 @@ var sub=func(a,b) {
     return res;
 }
 
+var hardamard=func(a,b) {
+    if(a.width!=b.width or a.height!=b.height) {
+        return nil;
+    }
+
+    var res=mat(a.width,a.height);
+    var (width,height,ref)=(res.width,res.height,res.mat);
+    var (aref,bref)=(a.mat,b.mat);
+
+    for(var i=0;i<height;i+=1) {
+        for(var j=0;j<width;j+=1) {
+            ref[i*width+j]=aref[i*width+j]*bref[i*width+j];
+        }
+    }
+    return res;
+}
+
 var neg=func(a) {
     var res=mat(a.width,a.height);
     var (aref,ref)=(a.mat,res.mat);
@@ -73,6 +89,25 @@ var neg=func(a) {
     return res;
 }
 
+var sum=func(a) {
+    var res=0;
+    var aref=a.mat;
+    forindex(var i;aref) {
+        res+=aref[i];
+    }
+    return res;
+}
+
+var mult_num=func(a,c) {
+    var res=mat(a.width,a.height);
+    var ref=res.mat;
+    var aref=a.mat;
+    forindex(var i;aref) {
+        ref[i]=aref[i]*c;
+    }
+    return res;
+}
+
 var trans=func(a) {
     var res=mat(a.height,a.width);
     var ref=res.mat;
@@ -119,12 +154,22 @@ var sigmoid=func(x) {
     return 1/(1+t);
 }
 
+var diffsigmoid=func(x) {
+    x=sigmoid(x);
+    return x*(1-x);
+}
+
 var tanh=func(x) {
     var t1=math.exp(x);
     var t2=math.exp(-x);
     return (t1-t2)/(t1+t2);
 }
 
+var difftanh=func(x) {
+    x=tanh(x);
+    return 1-x*x;
+}
+
 var test=func() {
     for(var i=0;i<1e4;i+=1) {
         var a=mat(4,20);
@@ -145,4 +190,83 @@ var test=func() {
     }
 }
 
-test();
\ No newline at end of file
+var bp_example=func() {
+    srand();
+    var lr=0.01;
+    var input=[
+        {width:2,height:1,mat:[0,0]},
+        {width:2,height:1,mat:[0,1]},
+        {width:2,height:1,mat:[1,0]},
+        {width:2,height:1,mat:[1,1]}
+    ];
+    # last 2 column is useless, only used to make sure bp runs correctly
+    var expect=[
+        {width:3,height:1,mat:[0,0,0]},
+        {width:3,height:1,mat:[1,0,0]},
+        {width:3,height:1,mat:[1,0,0]},
+        {width:3,height:1,mat:[0,0,0]}
+    ];
+    var hidden={
+        weight:mat(4,2),
+        bias:mat(4,1),
+        in:nil,
+        out:nil,
+        diff:nil
+    };
+    var output={
+        weight:mat(3,4),
+        bias:mat(3,1),
+        in:nil,
+        out:nil,
+        diff:nil
+    };
+    rand_init(hidden.weight);
+    rand_init(hidden.bias);
+    rand_init(output.weight);
+    rand_init(output.bias);
+
+    var epoch=0;
+    var total=1e6;
+    while(total>0.01) {
+        epoch+=1;
+        if(epoch>1e4) {
+            println("Training failed after ",epoch," epoch.");
+            break;
+        }
+
+        total=0;
+        forindex(var i;input) {
+            hidden.in=add(mult(input[i],hidden.weight),hidden.bias);
+            hidden.out=activate(hidden.in,tanh);
+
+            output.in=add(mult(hidden.out,output.weight),output.bias);
+            output.out=activate(output.in,sigmoid);
+
+            var error=sub(expect[i],output.out);
+
+            output.diff=hardamard(error,activate(output.in,diffsigmoid));
+            hidden.diff=hardamard(trans(mult(output.weight,trans(output.diff))),activate(hidden.in,difftanh));
+
+            output.bias=add(output.bias,output.diff);
+            hidden.bias=add(hidden.bias,hidden.diff);
+
+            output.weight=add(output.weight,mult(trans(hidden.out),output.diff));
+            hidden.weight=add(hidden.weight,mult(trans(input[i]),hidden.diff));
+
+            total+=sum(mult_num(mult(error,trans(error)),0.5));
+        }
+    }
+    if(epoch<=1e4) {
+        println("Training succeeded after ",epoch," epoch.");
+    }
+
+    forindex(var i;input) {
+        hidden.in=add(mult(input[i],hidden.weight),hidden.bias);
+        hidden.out=activate(hidden.in,tanh);
+
+        output.in=add(mult(hidden.out,output.weight),output.bias);
+        output.out=activate(output.in,sigmoid);
+
+        println(input[i].mat," : ",output.out.mat);
+    }
+}
\ No newline at end of file
diff --git a/test/md5compare.nas b/test/md5compare.nas
index 175c497..a1e4900 100644
--- a/test/md5compare.nas
+++ b/test/md5compare.nas
@@ -33,9 +33,9 @@ var compare=func(){
 var filechecksum=func(){
     var files=[
         "./stl/fg_env.nas",         "./stl/file.nas",
-        "./stl/json.nas",
-        "./stl/lib.nas",            "./stl/list.nas",
-        "./stl/log.nas",            "./stl/module.nas",
+        "./stl/json.nas",           "./stl/lib.nas",
+        "./stl/list.nas",           "./stl/log.nas",
+        "./stl/mat.nas",            "./stl/module.nas",
         "./stl/padding.nas",        "./stl/process_bar.nas",
         "./stl/queue.nas",          "./stl/result.nas",
         "./stl/sort.nas",           "./stl/stack.nas",