primrules.py

# Copyright (c) 2022 PaddlePaddle Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
import typing
import math

import paddle

from . import primops
from .primops import (add, broadcast, concat, cos, div, exp, fill_const, gather,
                      matmul, mul, neg, reduce, reshape, scatter_add, set_value,
                      sin, slice_assign, slice_select, split, sqrt, sub, tanh,
                      transpose, log, select, eq, max, erf)
from .primreg import (REGISTER_JVP, REGISTER_ORIG2PRIM, REGISTER_PRIM2ORIG,
                      REGISTER_TRANSPOSE, lookup_fn, lookup_jvp,
                      lookup_orig2prim, lookup_prim2orig, lookup_transpose,
                      op_position_inputs, op_position_output)
from .utils import INT_DTYPE_2_STRING, get_input_var_list, get_output_var_list


def _orig2prim(op, *args):
    _lowerrule = lookup_orig2prim(op.type)
    return _lowerrule(op, *args)


def _prim2orig(op, *args):
    _lowerrule = lookup_prim2orig(op.type)
    return _lowerrule(op, *args)


def _jvp(op, *args):
    _jvprule = lookup_jvp(op.type)
    return _jvprule(op, *args)


def _transpose(op, dot_checker, *args):
    _transposerule = lookup_transpose(op.type)
    return _transposerule(op, dot_checker, *args)


def linear_jvp(op, *args, **kwargs):
    fn = lookup_fn(op.type)
    out_dot = fn(*args, **kwargs)
    return out_dot


## Register orig2prim lower rules
"""
These original ops are fully supported:

elementwise_add
elementwise_sub
elementwise_mul
tanh
fill_zeros_like
sum
index_select
scale
assign
sqrt
log
select
equal
elementwise_pow

These original ops are partially supported:

matmul_v2
reshape2
concat
slice
p_norm
"""


@REGISTER_ORIG2PRIM('elementwise_add')
def elementwise_add_orig2prim(op, x, y):
    if x.shape != y.shape:
        y = broadcast(y, shape=x.shape)
    if op.attr('Scale_x') - 1.0 > 1e-5:
        scale_x = fill_const(shape=x.shape,
                             dtype=x.dtype,
                             value=op.attr('Scale_x'))
        x = mul(x, scale_x)
    if op.attr('Scale_y') - 1.0 > 1e-5:
        scale_y = fill_const(shape=y.shape,
                             dtype=y.dtype,
                             value=op.attr('Scale_y'))
        y = mul(y, scale_y)
    z = add(x, y)
    if op.attr('Scale_out') - 1.0 > 1e-5:
        scale_out = fill_const(shape=z.shape,
                               dtype=z.dtype,
                               value=op.attr('Scale_out'))
        z = mul(z, scale_out)
    return z


@REGISTER_ORIG2PRIM('elementwise_sub')
def elementwise_sub_orig2prim(op, x, y):
    if x.shape != y.shape:
        y = broadcast(y, shape=x.shape)
    if op.attr('Scale_x') - 1.0 > 1e-5:
        scale_x = fill_const(shape=x.shape,
                             dtype=x.dtype,
                             value=op.attr('Scale_x'))
        x = mul(x, scale_x)
    if op.attr('Scale_y') - 1.0 > 1e-5:
        scale_y = fill_const(shape=y.shape,
                             dtype=y.dtype,
                             value=op.attr('Scale_y'))
        y = mul(y, scale_y)
    z = sub(x, y)
    if op.attr('Scale_out') - 1.0 > 1e-5:
        scale_out = fill_const(shape=z.shape,
                               dtype=z.dtype,
                               value=op.attr('Scale_out'))
        z = mul(z, scale_out)
    return z


@REGISTER_ORIG2PRIM('elementwise_mul')
def elementwise_mul_orig2prim(op, x, y):
    if x.shape != y.shape:
        y = broadcast(y, shape=x.shape)
    if op.attr('Scale_x') - 1.0 > 1e-5:
        scale_x = fill_const(shape=x.shape,
                             dtype=x.dtype,
                             value=op.attr('Scale_x'))
        x = mul(x, scale_x)
    if op.attr('Scale_y') - 1.0 > 1e-5:
        scale_y = fill_const(shape=y.shape,
                             dtype=y.dtype,
                             value=op.attr('Scale_y'))
        y = mul(y, scale_y)
    z = mul(x, y)
    if op.attr('Scale_out') - 1.0 > 1e-5:
        scale_out = fill_const(shape=z.shape,
                               dtype=z.dtype,
                               value=op.attr('Scale_out'))
        z = mul(z, scale_out)
    return z


@REGISTER_ORIG2PRIM('tanh')
def tanh_orig2prim(op, x):
    return tanh(x)


@REGISTER_ORIG2PRIM('sin')
def sin_orig2prim(op, x):
    return sin(x)


@REGISTER_ORIG2PRIM('cos')
def cos_orig2prim(op, x):
    return cos(x)


@REGISTER_ORIG2PRIM('exp')
def exp_orig2prim(op, x):
    return exp(x)


@REGISTER_ORIG2PRIM('erf')
def erf_orig2prim(op, x):
    return erf(x)


@REGISTER_ORIG2PRIM('log')
def log_orig2prim(op, x):
    return log(x)


@REGISTER_ORIG2PRIM('fill_zeros_like')
def fill_zeros_like_orig2prim(op, x):
    return fill_const(value=0.0, shape=x.shape, dtype=x.dtype)


@REGISTER_ORIG2PRIM('sum')
def sum_orig2prim(op, xs):
    x0 = xs[0]
    for x in xs[1:]:
        x0 = add(x0, x)
    return x0


@REGISTER_ORIG2PRIM('index_select')
def index_select_orig2prim(op, index_t, x):
    return gather(x, indextensor=index_t, axis=op.attr('dim'))


@REGISTER_ORIG2PRIM('scale')
def scale_orig2prim(op, scale_t, x):
    if scale_t is None:
        scale_t = fill_const(shape=x.shape,
                             dtype=x.dtype,
                             value=op.attr('scale'))
    bias_t = fill_const(shape=x.shape, dtype=x.dtype, value=op.attr('bias'))
    if op.attr('bias_after_scale'):
        return add(mul(x, scale_t), bias_t)
    else:
        return mul(add(x, bias_t), scale_t)


@REGISTER_ORIG2PRIM('assign')
def assign_orig2prim(op, x):
    zero_t = fill_const(shape=x.shape, dtype=x.dtype, value=0.0)
    return add(x, zero_t)


@REGISTER_ORIG2PRIM('sqrt')
def sqrt_orig2prim(op, x):
    return sqrt(x)


@REGISTER_ORIG2PRIM('matmul_v2')
def matmul_v2_orig2prim(op, x, y):

    def trans(shape):
        ret = [i for i in range(len(shape))]
        ret[-1], ret[-2] = ret[-2], ret[-1]
        return ret

    assert len(x.shape) < 4 and len(
        y.shape) < 4, 'Do not support multi batchsize dimensions currently.'

    if len(x.shape) == 1:
        x = broadcast(x, shape=[1, x.shape[0]])
    if len(y.shape) == 1:
        y = broadcast(y, shape=[y.shape[0], 1])
    if op.attr('trans_x'):
        x = transpose(x, axis=trans(x.shape))
    if op.attr('trans_y'):
        y = transpose(y, axis=trans(y.shape))
    return matmul(x, y)


## NOTE(lml): The second output of reshape2 Xshape, which is only used in reshape2_grad, is meanlingless in new autograd mechanism, thus we use a zero tensor instead.
@REGISTER_ORIG2PRIM('reshape2')
def reshape2_orig2prim(op, shape_t, shape_tl, x):
    assert shape_t is None, 'Can not lower reshape2 into prim ops with shapetensor.'
    assert shape_tl is None, 'Can not lower reshape2 into prim ops with shapetensorlist.'
    y, xshape = get_output_var_list(op)
    return reshape(x, shape=y.shape), fill_const(shape=xshape.shape,
                                                 dtype=xshape.dtype,
                                                 value=0.0)


@REGISTER_ORIG2PRIM('concat')
def concat_orig2prim(op, axis_t, xs):
    assert axis_t is None, 'Can not lower concat into prim ops with axistensor.'
    return concat(xs, axis=op.attr('axis'))


@REGISTER_ORIG2PRIM('slice')
def slice_orig2prim(op, ends_t, ends_tl, x, starts_t, starts_tl):
    assert starts_t is None, 'Can not lower concat into prim ops with startstensor.'
    assert ends_t is None, 'Can not lower concat into prim ops with endstensor.'
    assert starts_tl is None, 'Can not lower concat into prim ops with startstensorlist.'
    assert ends_tl is None, 'Can not lower concat into prim ops with endstensorlist.'
    starts = op.attr('starts')
    ends = op.attr('ends')
    strides = [1 for _ in starts]
    axis = op.attr('axes')
    y = slice_select(x, starts=starts, ends=ends, strides=strides, axis=axis)
    if op.attr('decrease_axis'):
        y = reshape(y, shape=get_output_var_list(op)[0].shape)
    return y


@REGISTER_ORIG2PRIM('p_norm')
def p_norm_orig2prim(op, x):

    def num_el(shape):
        n = 1
        for s in shape:
            n = n * s
        return n

    assert op.attr(
        'asvector'), 'Only support lower pnorm when asvector=True currently'
    if len(x.shape) > 1:
        x = reshape(x, shape=[num_el(x.shape)])

    if abs(op.attr('porder') - 2.0) < 1e-5:
        return sqrt(reduce(mul(x, x), axis=[0]))
    elif abs(op.attr('porder') - 1.0) < 1e-5:
        return reduce(sqrt(mul(x, x)), axis=[0])
    else:
        raise RuntimeError('Only support lower l2/l1 norm currently')


# TODO: support broadcast
@REGISTER_ORIG2PRIM('where')
def select_orig2prim(op, condition, x, y):
    return select(condition, x, y)


@REGISTER_ORIG2PRIM('equal')
def equal_orig2prim(op, x, y):
    if x.shape != y.shape:
        y = broadcast(y, shape=x.shape)
    return eq(x, y)


@REGISTER_ORIG2PRIM('elementwise_pow')
def elementwise_pow_orig2prim(op, x, y):
    if x.shape != y.shape:
        y = broadcast(y, shape=x.shape)

    z = primops.pow(x, y)
    return z


@REGISTER_ORIG2PRIM('elementwise_max')
def elementwise_max_orig2prim(op, x, y):
    if x.shape != y.shape:
        y = broadcast(y, shape=x.shape)
    return primops.max(x, y)


@REGISTER_ORIG2PRIM('gelu')
def gelu_orig2prim(op, x):
    if op.attr('approximate'):
        cdf = mul(
            fill_const(0.5, x.shape, x.dtype),
            add(
                fill_const(1.0, x.shape, x.dtype),
                tanh(
                    mul(
                        fill_const(math.sqrt(2 / math.pi), x.shape, x.dtype),
                        add(
                            x,
                            mul(
                                fill_const(0.044715, x.shape, x.dtype),
                                primops.pow(x, fill_const(3., x.shape,
                                                          x.dtype))))))))
        return mul(x, cdf)
    else:
        return mul(
            mul(fill_const(0.5, x.shape, x.dtype), x),
            add(fill_const(1.0, x.shape, x.dtype),
                erf(mul(x, fill_const(1 / math.sqrt(2.), x.shape, x.dtype)))))


## Register prim2orig lower rules
@REGISTER_PRIM2ORIG('add_p')
def add_prim2orig(op, x, y):
    return paddle.add(x, y)


@REGISTER_PRIM2ORIG('sub_p')
def sub_prim2orig(op, x, y):
    return paddle.subtract(x, y)


@REGISTER_PRIM2ORIG('mul_p')
def mul_prim2orig(op, x, y):
    return paddle.multiply(x, y)


@REGISTER_PRIM2ORIG('div_p')
def div_prim2orig(op, x, y):
    return paddle.divide(x, y)


@REGISTER_PRIM2ORIG('sqrt_p')
def sqrt_prim2orig(op, x):
    return paddle.sqrt(x)


@REGISTER_PRIM2ORIG('tanh_p')
def tanh_prim2orig(op, x):
    return paddle.tanh(x)


@REGISTER_PRIM2ORIG('sin_p')
def sin_prim2orig(op, x):
    return paddle.sin(x)


@REGISTER_PRIM2ORIG('cos_p')
def cos_prim2orig(op, x):
    return paddle.cos(x)


@REGISTER_PRIM2ORIG('exp_p')
def exp_prim2orig(op, x):
    return paddle.exp(x)


@REGISTER_PRIM2ORIG('erf_p')
def erf_prim2orig(op, x):
    return paddle.erf(x)


@REGISTER_PRIM2ORIG('log_p')
def log_prim2orig(op, x):
    return paddle.log(x)


@REGISTER_PRIM2ORIG('reshape_p')
def reshape_prim2orig(op, x):
    return paddle.reshape(x, shape=op.attr('shape'))


@REGISTER_PRIM2ORIG('broadcast_p')
def broadcast_prim2orig(op, x):
    return paddle.broadcast_to(x, shape=op.attr('shape'))


@REGISTER_PRIM2ORIG('transpose_p')
def transpose_prim2orig(op, x):
    return paddle.transpose(x, perm=op.attr('axis'))


@REGISTER_PRIM2ORIG('split_p')
def split_prim2orig(op, x):
    num_or_sections = op.attr('num_or_sections')
    if len(num_or_sections) == 1:
        num_or_sections = num_or_sections[0]
    return paddle.split(x,
                        num_or_sections=num_or_sections,
                        axis=op.attr('axis'))


@REGISTER_PRIM2ORIG('concat_p')
def concat_prim2orig(op, xs):
    return paddle.concat(xs, axis=op.attr('axis'))


@REGISTER_PRIM2ORIG('reduce_p')
def reduce_prim2orig(op, x):
    return paddle.sum(x, axis=op.attr('axis'), keepdim=op.attr('keepdim'))


@REGISTER_PRIM2ORIG('matmul_p')
def matmul_prim2orig(op, x, y):
    return paddle.matmul(x, y)


@REGISTER_PRIM2ORIG('slice_select_p')
def slice_select_prim2orig(op, x):
    return paddle.strided_slice(x,
                                axes=op.attr('axis'),
                                starts=op.attr('starts'),
                                ends=op.attr('ends'),
                                strides=op.attr('strides'))


@REGISTER_PRIM2ORIG('slice_assign_p')
def slice_assign_prim2orig(op, x, y):
    x_copy = paddle.assign(x)
    return set_value(x_copy,
                     y,
                     axis=op.attr('axis'),
                     starts=op.attr('starts'),
                     ends=op.attr('ends'),
                     strides=op.attr('strides'),
                     out=x_copy)


@REGISTER_PRIM2ORIG('gather_p')
def gather_prim2orig(op, index_t, x):
    return paddle.gather(x, index_t, axis=op.attr('axis'))


@REGISTER_PRIM2ORIG('scatter_add_p')
def scatter_add_prim2orig(op, index_t, x, y):
    assert op.attr('axis') == 0, 'Only support axis==0 currently'
    zeros = paddle.zeros_like(x=x, dtype=x.dtype)
    tmp = paddle.scatter(x=zeros, index=index_t, updates=y, overwrite=False)
    return paddle.add(x, tmp)


@REGISTER_PRIM2ORIG('fill_constant_p')
def fill_constant_prim2orig(op):
    return paddle.full(shape=op.attr('shape'),
                       fill_value=op.attr('value'),
                       dtype=INT_DTYPE_2_STRING[op.attr('dtype')])


@REGISTER_PRIM2ORIG('select_p')
def select_prim2orig(op, condition, x, y):
    return paddle.where(condition, x, y)


@REGISTER_PRIM2ORIG('eq_p')
def eq_prim2orig(op, x, y):
    return paddle.equal(x, y)


@REGISTER_PRIM2ORIG('pow_p')
def pow_prim2orig(op, x, y):
    return paddle.pow(x, y)


@REGISTER_PRIM2ORIG('max_p')
def max_prim2orig(op, x, y):
    return paddle.maximum(x, y)


## Register linearize rules
@REGISTER_JVP('add_p')
def add_jvp(op, x_dot, y_dot):
    if x_dot is None:
        return y_dot
    elif y_dot is None:
        return x_dot
    else:
        return linear_jvp(op, x_dot, y_dot)


@REGISTER_JVP('sub_p')
def sub_jvp(op, x_dot, y_dot):
    if x_dot is None:
        return neg(y_dot)
    elif y_dot is None:
        return x_dot
    else:
        return linear_jvp(op, x_dot, y_dot)


@REGISTER_JVP('mul_p')
def mul_jvp(op, x_dot, y_dot):
    if x_dot is None and y_dot is None:
        return None
    x, y = op_position_inputs(op)
    if x_dot is None:
        return mul(x, y_dot)
    elif y_dot is None:
        return mul(x_dot, y)
    else:
        t1, t2 = mul(x_dot, y), mul(x, y_dot)
        z_dot = add(t1, t2)
        return z_dot


@REGISTER_JVP('div_p')
def div_jvp(op, x_dot, y_dot):
    if x_dot is None and y_dot is None:
        return None
    x, y = op_position_inputs(op)
    if y_dot is None:
        return div(x_dot, y)
    elif x_dot is None:
        return neg(div(mul(x, y_dot), mul(y, y)))
    else:
        t1 = div(x_dot, y)
        t2 = div(mul(x, y_dot), mul(y, y))
        return sub(t1, t2)


@REGISTER_JVP('sqrt_p')
def sqrt_jvp(op, x_dot):
    if x_dot is None:
        return None
    y = op_position_output(op)
    c2 = fill_const(value=2.0, shape=y.shape, dtype=y.dtype)
    y_dot = div(x_dot, mul(c2, y))
    return y_dot


@REGISTER_JVP('tanh_p')
def tanh_jvp(op, x_dot):
    if x_dot is None:
        return None
    y = op_position_output(op)
    c1 = fill_const(value=1.0, shape=y.shape, dtype=y.dtype)
    y_dot = mul(x_dot, sub(c1, mul(y, y)))
    return y_dot


@REGISTER_JVP('sin_p')
def sin_jvp(op, x_dot):
    if x_dot is None:
        return None
    x, = op_position_inputs(op)
    return mul(x_dot, cos(x))


@REGISTER_JVP('cos_p')
def cos_jvp(op, x_dot):
    if x_dot is None:
        return None
    x, = op_position_inputs(op)
    return mul(x_dot, neg(sin(x)))


@REGISTER_JVP('exp_p')
def exp_jvp(op, x_dot):
    if x_dot is None:
        return None
    y = op_position_output(op)
    return mul(x_dot, y)


@REGISTER_JVP('erf_p')
def erf_jvp(op, x_dot):
    if x_dot is None:
        return None
    x, = op_position_inputs(op)
    return mul(
        fill_const(2. / math.sqrt(math.pi), x.shape, x.dtype),
        mul(x_dot, exp(neg(primops.pow(x, fill_const(2., x.shape, x.dtype))))))


@REGISTER_JVP('log_p')
def log_jvp(op, x_dot):
    if x_dot is None:
        return None
    x, = op_position_inputs(op)
    return div(x_dot, x)


@REGISTER_JVP('reshape_p')
def reshape_jvp(op, x_dot):
    if x_dot is None:
        return None
    shape = op.attr('shape')
    return linear_jvp(op, x_dot, shape=shape)


@REGISTER_JVP('broadcast_p')
def broadcast_jvp(op, x_dot):
    if x_dot is None:
        return None
    shape = op.attr('shape')
    return linear_jvp(op, x_dot, shape=shape)


@REGISTER_JVP('transpose_p')
def transpose_jvp(op, x_dot):
    if x_dot is None:
        return None
    axis = op.attr('axis')
    return linear_jvp(op, x_dot, axis=axis)


@REGISTER_JVP('split_p')
def split_jvp(op, x_dot):
    if x_dot is None:
        return None
    num_or_sections = op.attr('num_or_sections')
    axis = op.attr('axis')
    return linear_jvp(op, x_dot, num_or_sections=num_or_sections, axis=axis)


@REGISTER_JVP('concat_p')
def concat_jvp(op, xs_dot):
    if xs_dot is None:
        return None
    axis = op.attr('axis')
    return linear_jvp(op, xs_dot, axis=axis)


@REGISTER_JVP('reduce_p')
def reduce_jvp(op, x_dot):
    if x_dot is None:
        return None
    axis = op.attr('axis')
    keepdim = op.attr('keepdim')
    return linear_jvp(op, x_dot, axis=axis, keepdim=keepdim)


@REGISTER_JVP('matmul_p')
def matmul_jvp(op, x_dot, y_dot):
    if x_dot is None and y_dot is None:
        return None
    x, y = op_position_inputs(op)
    if x_dot is None:
        return matmul(x, y_dot)
    elif y_dot is None:
        return matmul(x_dot, y)
    else:
        t1 = matmul(x, y_dot)
        t2 = matmul(x_dot, y)
        return add(t1, t2)


@REGISTER_JVP('slice_select_p')
def slice_select_jvp(op, x_dot):
    if x_dot is None:
        return x_dot
    axis = op.attr('axis')
    starts = op.attr('starts')
    ends = op.attr('ends')
    strides = op.attr('strides')
    return linear_jvp(op,
                      x_dot,
                      axis=axis,
                      starts=starts,
                      ends=ends,
                      strides=strides)


@REGISTER_JVP('slice_assign_p')
def slice_assign_jvp(op, x_dot, y_dot):
    if x_dot is None:
        assert y_dot is None, 'y_dot must be None.'
        return None
    else:
        assert y_dot is not None, 'y_dot should not be None.'
    axis = op.attr('axis')
    starts = op.attr('starts')
    ends = op.attr('ends')
    strides = op.attr('strides')
    return linear_jvp(op,
                      x_dot,
                      y_dot,
                      axis=axis,
                      starts=starts,
                      ends=ends,
                      strides=strides)


@REGISTER_JVP('gather_p')
def gather_jvp(op, x_dot, indextensor):
    if x_dot is None:
        return None
    _, indextensor = op_position_inputs(op)
    axis = op.attr('axis')
    return linear_jvp(op, x_dot, indextensor, axis=axis)


@REGISTER_JVP('scatter_add_p')
def scatter_add_jvp(op, x_dot, y_dot):
    if x_dot is None:
        return None
    _, _, indextensor = op_position_inputs(op)
    axis = op.attr('axis')
    return linear_jvp(op, x_dot, y_dot, indextensor, axis=axis)


@REGISTER_JVP('select_p')
def select_jvp(op, cond_dot, x_dot, y_dot):
    if x_dot is None and y_dot is None:
        return None

    cond, x, y = op_position_inputs(op)
    if x_dot is None:
        x_dot = fill_const(value=0.0, shape=y.shape, dtype=y.dtype)
    if y_dot is None:
        y_dot = fill_const(value=0.0, shape=y.shape, dtype=y.dtype)
    return select(cond, x_dot, y_dot)


@REGISTER_JVP('eq_p')
def eq_jvp(op, x_dot, y_dot):
    if x_dot is None and y_dot is None:
        return None
    x, _ = op_position_inputs(op)
    z_dot = fill_const(value=0., shape=x.shape, dtype=x.dtype)
    return z_dot


@REGISTER_JVP('pow_p')
def pow_jvp(op, x_dot, y_dot):

    def _compute_t1(x, y):
        zero_y = fill_const(value=0.0, shape=y.shape, dtype=y.dtype)
        one_y = fill_const(value=1.0, shape=y.shape, dtype=y.dtype)

        cond = eq(y, zero_y)
        new_y = select(cond, one_y, sub(y, one_y))
        t1 = mul(x_dot, mul(y, primops.pow(x, new_y)))
        return t1

    if x_dot is None and y_dot is None:
        return None
    x, y = op_position_inputs(op)
    z = op_position_output(op)

    if y_dot is None:
        return _compute_t1(x, y)
    elif x_dot is None:
        return mul(y_dot, mul(log(x), z))
    else:
        t1, t2 = _compute_t1(x, y), mul(y_dot, mul(log(x), z))
        z_dot = add(t1, t2)
        return z_dot


@REGISTER_JVP('max_p')
def max_jvp(op, x_dot, y_dot):
    if x_dot is None and y_dot is None:
        return None

    x, y = op_position_inputs(op)
    z = op_position_output(op)
    z_zeros = fill_const(value=0.0, shape=z.shape, dtype=z.dtype)

    # To make the grad of max_p consistent with paddle.maximum when x==y,
    # we just let z_dot = y_dot when compute z_dot to y and x==y,
    # instead of using balance_eq like Jax.
    if y_dot is None:
        return select(eq(y, z), z_zeros, x_dot)
    elif x_dot is None:
        return select(eq(y, z), y_dot, z_zeros)
    else:
        return select(eq(y, z), y_dot, x_dot)


## Register transpose rules


@REGISTER_TRANSPOSE('add_p')
def add_transpose(op, check_dot, z_bar):
    x, y = op_position_inputs(op)
    assert check_dot(x) or check_dot(y), (
        f'(check_dot(x) or check_dot(y)) must be True, '
        f'but check_dot(x)={check_dot(x)} and check_dot(y)={check_dot(y)}.')
    x_bar = z_bar if check_dot(x) else None
    y_bar = z_bar if check_dot(y) else None
    return x_bar, y_bar


@REGISTER_TRANSPOSE('sub_p')
def sub_transpose(op, check_dot, z_bar):
    x, y = op_position_inputs(op)
    assert check_dot(x) or check_dot(y), (
        f'(check_dot(x) or check_dot(y)) must be True, '
        f'but check_dot(x)={check_dot(x)} and check_dot(y)={check_dot(y)}.')
    x_bar = z_bar if check_dot(x) else None
    y_bar = neg(z_bar) if check_dot(y) else None
    return x_bar, y_bar


@REGISTER_TRANSPOSE('mul_p')
def mul_transpose(op, check_dot, z_bar):
    x, y = op_position_inputs(op)
    assert check_dot(x) ^ check_dot(y), (
        f'(check_dot(x) ^ check_dot(y)) must be True, '
        f'but check_dot(x)={check_dot(x)} and check_dot(y)={check_dot(y)}.')
    if check_dot(x):
        return mul(z_bar, y), None
    else:
        return None, mul(x, z_bar)


@REGISTER_TRANSPOSE('div_p')
def div_transpose(op, check_dot, z_bar):
    x, y = op_position_inputs(op)
    assert not check_dot(y), 'check_dot(y) must be False'
    x_bar = div(z_bar, y) if check_dot(x) else None
    return x_bar, None


@REGISTER_TRANSPOSE('reshape_p')
def reshape_transpose(op, check_dot, y_bar):
    x, = op_position_inputs(op)
    assert check_dot(x), 'check_dot(x) must be True'
    return reshape(y_bar, shape=x.shape)


@REGISTER_TRANSPOSE('broadcast_p')
def broadcast_transpose(op, check_dot, y_bar):
    x, = op_position_inputs(op)
    assert check_dot(x), 'check_dot(x) must be True'
    bat = len(y_bar.shape) - len(x.shape)
    axis = list(range(bat))
    keepdim = [(bat + i) for i, s in enumerate(x.shape) if s == 1]
    axis += keepdim
    # TODO: Change it. keepdim boolean
    out = reduce(y_bar, axis=axis, keepdim=False)
    return reshape(out, x.shape)


@REGISTER_TRANSPOSE('transpose_p')
def transpose_transpose(op, check_dot, y_bar):
    x, = op_position_inputs(op)
    assert check_dot(x), 'check_dot(x) must be True'
    axis = op.attr('axis')
    reordered = sorted((k, i) for i, k in enumerate(axis))
    axis = [i for k, i in reordered]
    return transpose(y_bar, axis=axis)


@REGISTER_TRANSPOSE('split_p')
def split_transpose(op, check_dot, ys_bar):
    x, = op_position_inputs(op)
    assert check_dot(x), 'check_dot(x) must be True'
    return concat(ys_bar, axis=op.attr('axis'))


@REGISTER_TRANSPOSE('concat_p')
def concat_transpose(op, check_dot, y_bar):
    xs, = op_position_inputs(op)
    if not isinstance(xs, typing.Sequence):
        xs = [xs]
    for x in xs:
        assert check_dot(x), 'check_dot(x) must be True'
    axis = op.attr('axis')
    sections = [x.shape[axis] for x in xs]
    if len(sections) == 1:
        return y_bar
    return split(y_bar, num_or_sections=sections, axis=axis)


@REGISTER_TRANSPOSE('reduce_p')
def reduce_transpose(op, check_dot, y_bar):
    x, = op_position_inputs(op)
    assert check_dot(x), 'check_dot(x) must be True'
    axes = op.attr('axis')
    shape = tuple(1 if i in axes else size for i, size in enumerate(x.shape))
    t = reshape(y_bar, shape=shape)
    return broadcast(t, shape=x.shape)


@REGISTER_TRANSPOSE('matmul_p')
def matmul_transpose(op, check_dot, z_bar):
    x, y = op_position_inputs(op)
    assert check_dot(x) ^ check_dot(y), (
        f'(check_dot(x) ^ check_dot(y)) must be True, '
        f'but check_dot(x)={check_dot(x)} and check_dot(y)={check_dot(y)}.')
    # TODO: replace it. this is hacky
    axis = [1, 0] if len(x.shape) == 2 else [0, 2, 1]
    if check_dot(x):
        return matmul(z_bar, transpose(y, axis=axis)), None
    else:
        return None, matmul(transpose(x, axis=axis), z_bar)


@REGISTER_TRANSPOSE('slice_select_p')
def slice_select_transpose(op, check_dot, y_bar):
    x, = op_position_inputs(op)
    assert check_dot(x), 'check_dot(x) must be True'
    zeros = fill_const(value=0.0, shape=x.shape, dtype=x.dtype)
    axis = op.attr('axis')
    starts = op.attr('starts')
    ends = op.attr('ends')
    strides = op.attr('strides')
    return slice_assign(zeros,
                        y_bar,
                        axis=axis,
                        starts=starts,
                        ends=ends,
                        strides=strides)


@REGISTER_TRANSPOSE('slice_assign_p')
def slice_assign_transpose(op, check_dot, z_bar):
    x, y = op_position_inputs(op)
    assert check_dot(x) and check_dot(y), (
        f'(check_dot(x) and check_dot(y)) must be True, '
        f'but check_dot(x)={check_dot(x)} and check_dot(y)={check_dot(y)}.')
    zeros = fill_const(value=0.0, shape=y.shape, dtype=y.dtype)
    axis = op.attr('axis')
    starts = op.attr('starts')
    ends = op.attr('ends')
    strides = op.attr('strides')
    x_bar = slice_assign(z_bar,
                         zeros,
                         axis=axis,
                         starts=starts,
                         ends=ends,
                         strides=strides)
    y_bar = slice_select(z_bar,
                         axis=axis,
                         starts=starts,
                         ends=ends,
                         strides=strides)
    return x_bar, y_bar


@REGISTER_TRANSPOSE('gather_p')
def gather_transpose(op, check_dot, y_bar):
    x, indextensor = op_position_inputs(op)
    assert check_dot(x), 'check_dot(x) must be True'
    axis = op.attr('axis')
    zeros = fill_const(0.0, x.shape, x.dtype)
    x_bar = scatter_add(zeros, y_bar, indextensor, axis=axis)
    indextensor_bar = None
    return x_bar, indextensor_bar


@REGISTER_TRANSPOSE('scatter_add_p')
def scatter_add_transpose(op, check_dot, z_bar):
    x, y, indextensor = op_position_inputs(op)
    assert check_dot(x) and check_dot(y), (
        f'(check_dot(x) and check_dot(y)) must be True, '
        f'but check_dot(x)={check_dot(x)} and check_dot(y)={check_dot(y)}.')
    axis = op.attr('axis')
    zeros = fill_const(value=0.0, shape=y.shape, dtype=y.dtype)
    x_bar = scatter_add(z_bar, zeros, indextensor, axis=axis)
    y_bar = gather(z_bar, indextensor, axis=axis)
    indextensor_bar = None
    return x_bar, y_bar, indextensor_bar


@REGISTER_TRANSPOSE('select_p')
def select_transpose(op, check_dot, z_bar):
    cond, x, y = op_position_inputs(op)
    assert check_dot(cond) or check_dot(x) or check_dot(y), (
        f'check_dot(cond) ^ (check_dot(x) ^ check_dot(y)) must be True, '
        f'but check_dot(cond)={check_dot(cond)}, check_dot(x)={check_dot(x)} and check_dot(y)={check_dot(y)}.'
    )

    zeros_x = fill_const(value=0.0, shape=x.shape, dtype=x.dtype)
    zeros_y = fill_const(value=0.0, shape=y.shape, dtype=y.dtype)

    cond_bar = fill_const(value=0.0, shape=y.shape,
                          dtype=cond.dtype) if check_dot(cond) else None
    x_bar = select(cond, z_bar, zeros_x) if check_dot(x) else None
    y_bar = select(cond, zeros_y, z_bar) if check_dot(y) else None

    return cond_bar, x_bar, y_bar