[prim] simplify batch_norm composite rule (#51827)

* simplify batch_norm composite rule

* polish code

paddle/fluid/prim/api/composite_backward/composite_backward_api.h

@@ -1192,7 +1192,7 @@ void batch_norm_grad(const Tensor& x,
     auto eps =
         full<T>(phi::vectorize(run_var.dims()), epsilon, run_var.dtype());
     mean_data = run_mean;
-    rsqrt_var = 1 / (run_var + eps).pow(0.5);
+    rsqrt_var = (run_var + eps).pow(-0.5);
   } else {
     mean_data = saved_mean;
     rsqrt_var = saved_variance;

python/paddle/incubate/autograd/composite_rules.py

@@ -71,7 +71,11 @@ def composite_batchnorm(
     use_global_stats,
     trainable_statistics,
 ):
-    """define composite rule of op batch_norm"""
+    """
+    define composite rule of op batch_norm
+    As the same with op kernel, the position of savedvariance indeed return inverse std.
+    """
+
     is_amp = False
     from paddle.fluid.data_feeder import convert_dtype
 
@@ -94,38 +98,42 @@ def composite_batchnorm(
         1 if i in reduce_axes else s for i, s in enumerate(x.shape)
     )
 
-    batch_mean = zeros(run_mean.shape, run_mean.dtype)
-    batch_var = zeros(run_var.shape, run_var.dtype)
-    if not use_run_stat:
+    half = -0.5
 
-        batch_mean = mean(x, reduce_axes, keepdim=True)
-        temp = mean(x * x, reduce_axes, keepdim=True)
+    if not use_run_stat:
+        batch_mean = mean(x, reduce_axes)
+        temp = mean(x * x, reduce_axes)
         batch_var = temp - batch_mean * batch_mean
+        inv_std = pow((batch_var + epsilon), half)
+        if data_layout == "NHWC":
+            x_hat = (x - batch_mean) * inv_std
+        else:
+            x_hat = (x - reshape(batch_mean, stats_shape)) * reshape(
+                inv_std, stats_shape
+            )
 
-        x_hat = (x - reshape(batch_mean, stats_shape)) / sqrt(
-            reshape(batch_var, stats_shape) + epsilon
-        )
-
-        run_mean = momentum * run_mean + (1 - momentum) * reshape(
-            batch_mean, run_mean.shape
-        )
-        run_var = momentum * run_var + (1 - momentum) * reshape(
-            batch_var, run_var.shape
-        )
+        run_mean = momentum * run_mean + (1 - momentum) * batch_mean
+        run_var = momentum * run_var + (1 - momentum) * batch_var
     else:
-        x_hat = (x - reshape(run_mean, stats_shape)) / sqrt(
-            reshape(run_var, stats_shape) + epsilon
-        )
-    y = reshape(scale, stats_shape) * x_hat + reshape(bias, stats_shape)
+        batch_mean = zeros(run_mean.shape, run_mean.dtype)
+        batch_var = zeros(run_var.shape, run_var.dtype)
+        inv_std = pow((batch_var + epsilon), half)
+        if data_layout == "NHWC":
+            x_hat = (x - run_mean) * pow((run_var + epsilon), half)
+        else:
+            x_hat = (x - reshape(run_mean, stats_shape)) * pow(
+                (reshape(run_var, stats_shape) + epsilon), half
+            )
+    if data_layout == "NHWC":
+        y = scale * x_hat + bias
+    else:
+        y = reshape(scale, stats_shape) * x_hat + reshape(bias, stats_shape)
     if is_amp:
         y = cast(y, "float16")
 
-    # As the same with op kernel, indeed return inverse std
-    inv_std = 1.0 / sqrt(batch_var + epsilon)
-
     # add op assign to detach tensor in void unsafe change outside the rule.
-    batch_mean_ = assign(reshape(batch_mean, run_mean.shape))
-    inv_std_ = assign(reshape(inv_std, run_var.shape))
+    batch_mean_ = assign(batch_mean)
+    inv_std_ = assign(inv_std)
     run_mean_ = assign(run_mean)
     run_var_ = assign(run_var)