lax_control_flow_test.py

# Copyright 2018 Google LLC
#
# 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
#
#     https://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 collections
from functools import partial
import itertools
import operator
import re
from unittest import SkipTest, skipIf
import textwrap

from absl.testing import absltest
from absl.testing import parameterized

import numpy as np
import numpy.random as npr

from jax import api
from jax import core
from jax import lax
from jax import random
from jax import test_util as jtu
from jax.util import unzip2
from jax.lib import xla_bridge
from jax.interpreters import xla
import jax.numpy as jnp  # scan tests use numpy
import jax.scipy as jsp

from jax.config import config
config.parse_flags_with_absl()


# Some tests are useful for testing both lax.cond and lax.switch. This function
# provides a lax.cond-compatible interface to a two-branch lax.switch. Several
# tests in this file are parameterized such that they either call into lax.cond
# or into this function.
def cond_via_switch(pred, true_fun, false_fun, op, *args):
  if len(args) > 0:
    assert len(args) == 1
    true_op, _true_fun, false_op, _false_fun = true_fun, false_fun, op, args[0]
    op = (false_op, true_op)
    false_fun = lambda op: _false_fun(op[0])
    true_fun = lambda op: _true_fun(op[1])
  index = lax.convert_element_type(pred, np.int32)
  return lax.switch(index, [false_fun, true_fun], op)


COND_IMPLS = [
    (lax.cond, 'cond'),
    (cond_via_switch, 'switch'),
]


SCAN_IMPLS = [
    (lax.scan, 'unroll1'),
    (partial(lax.scan, unroll=2), 'unroll2'),
]


def while_loop_reference(cond, body, carry):
  while cond(carry):
    carry = body(carry)
  return carry


def scan_reference(f, init, xs):
  carry = init
  ys = []
  for x in xs:
    (carry, y) = f(carry, x)
    ys.append(lax.reshape(y, (1,) + np.shape(y)))
  ys = lax.concatenate(ys, 0)
  return carry, ys


def high_precision_dot(a, b):
  return lax.dot(a, b, precision=lax.Precision.HIGHEST)


def posify(matrix):
  return high_precision_dot(matrix, matrix.T.conj())


class LaxControlFlowTest(jtu.JaxTestCase):

  def testWhileWithTuple(self):
    limit = 10

    def loop_cond(state):
      pos, _ = state
      return lax.lt(pos, limit)

    def loop_body(state):
      pos, count = state
      return (lax.add(pos, 1), lax.add(count, 1))

    def loop(init):
      result = lax.while_loop(loop_cond, loop_body, (init, 0))
      _, count = result
      return count

    cloop = api.jit(loop)

    self.assertEqual(loop(2), limit - 2)
    self.assertEqual(cloop(2), limit - 2)
    self.assertEqual(cloop(2), limit - 2)
    self.assertEqual(cloop(3), limit - 3)

  def testWhileWithManyArgs(self):
    nargs = 256

    def loop_cond(state):
      return lax.lt(state[0], 2)

    def loop_body(state):
      return tuple(lax.add(s, 1) for s in state)

    _ = lax.while_loop(loop_cond, loop_body, (0,) * nargs)

  def testNestedWhile(self):

    def outer_loop(num):  # pylint: disable=missing-docstring
      def cond_fun(state):
        num, i, _ = state
        return lax.lt(i, num)

      def body_fun(state):
        num, i, count = state
        return (num, lax.add(i, 1), inner_loop(i, count))

      init_val = (num, 0, 0)
      _, i, count = lax.while_loop(cond_fun, body_fun, init_val)
      return (i, count)

    def inner_loop(i, count):  # pylint: disable=missing-docstring
      def cond_fun(state):
        i, j, _ = state
        return lax.le(j, i)

      def body_fun(state):
        i, j, count = state
        return (i, lax.add(j, 1), lax.add(count, 1))

      init_val = (i, 0, count)
      _, _, count = lax.while_loop(cond_fun, body_fun, init_val)
      return count

    cloop = api.jit(outer_loop)

    self.assertEqual(outer_loop(3), (3, 6))
    self.assertEqual(cloop(3), (3, 6))
    self.assertEqual(cloop(3), (3, 6))
    self.assertEqual(cloop(2), (2, 3))
    self.assertEqual(cloop(4), (4, 10))

  def testWhileWithClosure(self):

    def loop(init, local_limit, inc):

      def loop_cond(state):
        pos, _ = state
        return lax.lt(pos, local_limit)

      def loop_body(state):
        effect[0] = True
        pos, count = state
        return (lax.add(pos, 1), lax.add(count, inc))

      result = lax.while_loop(loop_cond, loop_body, (init, 0))
      _, count = result
      return count

    cloop = api.jit(loop)

    limit = 10
    effect = [False]
    self.assertEqual(loop(2, limit, 1), limit - 2)
    assert effect[0]
    effect[0] = False
    self.assertEqual(cloop(2, limit, 1), limit - 2)
    assert effect[0]
    effect[0] = False
    self.assertEqual(cloop(2, limit, 1), limit - 2)
    self.assertEqual(cloop(3, limit, 1), limit - 3)
    assert not effect[0]

  def testWhileWithClosureJit(self):

    def loop(init, local_limit, inc):

      def loop_cond(state):
        pos, _ = state
        return lax.lt(pos, local_limit)

      def loop_body(state):
        effect[0] = True
        pos, count = state
        f = lambda pos, inc: (lax.add(pos, 1), lax.add(count, inc))
        return api.jit(f)(pos, inc)

      result = lax.while_loop(loop_cond, loop_body, (init, 0))
      _, count = result
      return count

    cloop = api.jit(loop)

    limit = 10
    effect = [False]
    self.assertEqual(loop(2, limit, 1), limit - 2)
    assert effect[0]
    effect[0] = False
    self.assertEqual(cloop(2, limit, 1), limit - 2)
    assert effect[0]
    effect[0] = False
    self.assertEqual(cloop(2, limit, 1), limit - 2)
    self.assertEqual(cloop(3, limit, 1), limit - 3)
    assert not effect[0]

  def testWhileTypeErrors(self):
    """Test typing error messages for while."""
    with self.assertRaisesRegex(TypeError,
        re.escape("cond_fun must return a boolean scalar, but got pytree PyTreeDef(tuple, [*,*]).")):
      lax.while_loop(lambda c: (1., 1.), lambda c: c, 0.)
    with  self.assertRaisesRegex(TypeError,
        re.escape("cond_fun must return a boolean scalar, but got output type(s) [ShapedArray(float32[])].")):
      lax.while_loop(lambda c: np.float32(1.), lambda c: c, np.float32(0.))
    with self.assertRaisesRegex(TypeError,
        re.escape("body_fun output and input must have same type structure, got PyTreeDef(tuple, [*,*]) and *.")):
      lax.while_loop(lambda c: True, lambda c: (1., 1.), 0.)
    with self.assertRaisesWithLiteralMatch(TypeError,
        ("body_fun output and input must have identical types, got\n"
         "ShapedArray(bool[], weak_type=True)\n"
         "and\n"
         "ShapedArray(float32[]).")):
      lax.while_loop(lambda c: True, lambda c: True, np.float32(0.))

  def testNestedWhileWithDynamicUpdateSlice(self):
    num = 5

    def update_entry(arr, val, i, j):
      val = lax.reshape(val, [1, 1])
      return lax.dynamic_update_slice(arr, val, (i, j))

    def outer_loop(arr):  # pylint: disable=missing-docstring

      def cond_fun(state):
        i, num, _, _ = state
        return lax.lt(i, num)

      def body_fun(state):
        i, num, arr, out = state
        return (lax.add(i, 1), num, arr, inner_loop(i, arr, out))

      out = np.zeros(arr.shape, dtype=arr.dtype)
      init_val = (0, num, arr, out)
      _, _, _, out = lax.while_loop(cond_fun, body_fun, init_val)
      return out

    def inner_loop(i, arr, out):  # pylint: disable=missing-docstring

      def cond_fun(state):
        i, j, _, _ = state
        return lax.le(j, i)

      def body_fun(state):
        i, j, arr, out = state
        arr_i = lax.dynamic_index_in_dim(arr, i, 0, False)
        arr_i_j = lax.dynamic_index_in_dim(arr_i, j, 0, False)
        out = update_entry(out, arr_i_j, i, j)
        return (i, lax.add(j, 1), arr, out)

      init_val = (i, 0, arr, out)
      _, _, _, out = lax.while_loop(cond_fun, body_fun, init_val)
      return out

    cloop = api.jit(outer_loop)
    arr = npr.RandomState(0).randn(5, 5)
    self.assertAllClose(outer_loop(arr), np.tril(arr), check_dtypes=False)
    self.assertAllClose(cloop(arr), np.tril(arr), check_dtypes=False)
    self.assertAllClose(cloop(arr), np.tril(arr), check_dtypes=False)

  def testLoopWithConjunctionCondition(self):
    def sum_first_n(arr, num):  # pylint: disable=missing-docstring
      def cond_fun(state):
        arr, num, i, _ = state
        return lax.bitwise_and(lax.lt(i, num), lax.lt(i, arr.shape[0]))

      def body_fun(state):
        arr, num, i, total = state
        arr_i = lax.dynamic_index_in_dim(arr, i, 0, False)
        return (arr, num, lax.add(i, 1), lax.add(total, arr_i))

      init_val = (arr, num, 0, 0.)
      _, _, _, total = lax.while_loop(cond_fun, body_fun, init_val)
      return total

    cfun = api.jit(sum_first_n)
    x = npr.RandomState(0).randn(10).astype(jnp.float_)

    for num in [0, 5, 10, 15]:
      self.assertAllClose(sum_first_n(x, num), np.sum(x[:num]),
                          check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)

  def testWhileLoopBatched(self):
    def fun(x):
      return lax.while_loop(lambda x: x < 3, lambda x: x + 2, x)

    ans = api.vmap(fun)(np.array([0, 1, 2, 3]))
    expected = np.array([4, 3, 4, 3])
    self.assertAllClose(ans, expected, check_dtypes=False)

    fun = api.jit(fun)
    ans = api.vmap(fun)(np.array([0, 1, 2, 3]))
    expected = np.array([4, 3, 4, 3])
    self.assertAllClose(ans, expected, check_dtypes=False)

  def testWhileLoopCondConstsBatched(self):
    def fun(x, y):
      return lax.while_loop(lambda x: x < y, lambda x: x + 2, x)

    ans = api.vmap(fun, in_axes=(None, 0))(0, np.array([2, 3]))
    expected = np.array([2, 4])
    self.assertAllClose(ans, expected, check_dtypes=False)

  def testWhileLoopBodyConstsBatched(self):
    def fun(x, y):
      return lax.while_loop(lambda x: x < 3, lambda x: x + y, x)

    ans = api.vmap(fun, in_axes=(None, 0))(0, jnp.array([2, 3]))
    expected = np.array([4, 3])
    self.assertAllClose(ans, expected, check_dtypes=False)

  def testWhileLoopTupleBatched(self):
    def cond_fun(loop_carry):
      x, y = loop_carry
      return x + y < 5

    def body_fun(loop_carry):
      x, y = loop_carry
      x = x + 1
      return x, y

    def fun(x, y):
      return lax.while_loop(cond_fun, body_fun, (x, y))

    ans = api.vmap(fun)(np.array([0, 0]), np.array([1, 2]))
    expected = (np.array([4, 3]), np.array([1, 2]))
    self.assertAllClose(ans, expected, check_dtypes=False)

  def test_issue_3204(self):
    # Error during XLA code generation for vmap of nested loops
    def test(a, b):
      val = 0
      i = 0
      j = 0

      condfun_1 = lambda inp: inp[1] < a + 1
      condfun_2 = lambda inp: inp[2] < b + 1

      def bodyfun_1(inp):
        val, i, j = inp
        j = 0

        def bodyfun_2(inp):
          val, i, j = inp
          val += i + j
          j += 1
          return (val, i, j)

        result = lax.while_loop(condfun_2, bodyfun_2, (val, i, j))
        val = result[0]
        i += 1
        return (val, i, j)

      result = lax.while_loop(condfun_1, bodyfun_1, (val, i, j))
      return result[0]

    arr = np.arange(5)
    vmap_test = api.vmap(test, (0, 0))
    vmap_test(arr, arr)

  def testForiLoopErrors(self):
    """Test typing error messages for while."""
    with self.assertRaisesRegex(
      TypeError, "arguments to fori_loop must have equal types"):
      lax.fori_loop(np.int16(0), jnp.int32(10), (lambda i, c: c), jnp.float32(7))

  def testForiLoopBatched(self):
    def body_fun(i, loop_carry):
      x, y = loop_carry
      x = x + 1
      y = y + 2
      return x, y

    def fun(x):
      return lax.fori_loop(0, 10, body_fun, (x, 0))

    ans = api.vmap(fun)(np.array([0, 1]))
    expected = (np.array([10, 11]), np.array([20, 20]))
    self.assertAllClose(ans, expected, check_dtypes=False)

  def testForiLoopBatchedIssue1190(self):
    cond_fun = lambda carry: carry[0] < 4
    body_fun = lambda carry: (carry[0] + 1, carry[1] + 1)
    f = lambda x: lax.while_loop(cond_fun, body_fun, (0, x))
    jaxpr = api.make_jaxpr(api.vmap(f))(jnp.arange(3))
    eqn = jaxpr.jaxpr.eqns[0]
    self.assertIs(eqn.primitive, lax.while_p)
    self.assertEqual(eqn.params['cond_jaxpr'].in_avals[0].shape, ())

  def testForiLoopBasic(self):
    def body_fun(i, tot):
      return lax.add(tot, i)

    def count(num):
      return lax.fori_loop(0, num, body_fun, 0)

    self.assertEqual(count(2), 1)
    self.assertEqual(count(3), 3)
    self.assertEqual(count(4), 6)

    for args_maker in [lambda: [2], lambda: [3], lambda: [4]]:
      self._CompileAndCheck(count, args_maker)

  def testForiLoopClosure(self):
    def count(num):
      def body_fun(i, tot):
        return lax.add(num, lax.add(tot, i))
      return lax.fori_loop(0, num, body_fun, 0)

    cfun = api.jit(count)

    self.assertEqual(count(2), 1 + 2**2)
    self.assertEqual(count(2), cfun(2))
    self.assertEqual(count(3), 3 + 3**2)
    self.assertEqual(count(3), cfun(3))
    self.assertEqual(count(4), 6 + 4**2)
    self.assertEqual(count(4), cfun(4))

  def testForiLoopTupleState(self):
    def sum_first_n(arr, num):
      def body_fun(i, state):
        arr, total = state
        arr_i = lax.dynamic_index_in_dim(arr, i, 0, False)
        return (arr, lax.add(total, arr_i))

      init_val = (arr, 0.)
      _, total = lax.fori_loop(0, lax.min(arr.shape[0], num), body_fun,
                               init_val)
      return total

    cfun = api.jit(sum_first_n)
    x = npr.RandomState(0).randn(10).astype(jnp.float_)

    for num in [0, 5, 10, 15]:
      self.assertAllClose(sum_first_n(x, num), np.sum(x[:num]),
                          check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)

  def testForiLoopDictState(self):
    def sum_first_n(arr, num):
      def body_fun(i, state):
        arr, total = state['arr'], state['total']
        arr_i = lax.dynamic_index_in_dim(arr, i, 0, False)
        return {'arr': arr, 'total': lax.add(total, arr_i)}

      init_val = {'arr': arr, 'total': 0.}
      out_val = lax.fori_loop(0, lax.min(arr.shape[0], num), body_fun, init_val)
      return out_val['total']

    cfun = api.jit(sum_first_n)
    x = npr.RandomState(0).randn(10).astype(jnp.float_)

    for num in [0, 5, 10, 15]:
      self.assertAllClose(sum_first_n(x, num), np.sum(x[:num]),
                          check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)

  def testForiLoopEmptyTupleInState(self):
    def sum_first_n(arr, num):
      def body_fun(i, state):
        arr, total, _ = state
        arr_i = lax.dynamic_index_in_dim(arr, i, 0, False)
        return (arr, lax.add(total, arr_i), ())

      init_val = (arr, 0., ())
      _, tot, _ = lax.fori_loop(0, lax.min(arr.shape[0], num), body_fun, init_val)
      return tot

    cfun = api.jit(sum_first_n)
    x = npr.RandomState(0).randn(10).astype(jnp.float_)

    for num in [0, 5, 10, 15]:
      self.assertAllClose(sum_first_n(x, num), np.sum(x[:num]),
                          check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)
      self.assertAllClose(cfun(x, num), np.sum(x[:num]), check_dtypes=False)

  def testCond(self):
    def fun(x):
      if x < 3:
        return (x, x)
      else:
        y = lax.mul(2, x)
        return y, lax.mul(2, y)

    @api.jit
    def cfun(x):
      def false_fun(x):
        y = lax.mul(2, x)
        return y, lax.mul(2, y)
      return lax.cond(lax.lt(x, 3), lambda x: (x, x), false_fun, x)

    self.assertEqual(fun(0), cfun(0))
    self.assertEqual(fun(0), (0, 0))
    self.assertEqual(fun(1), cfun(1))
    self.assertEqual(fun(1), (1, 1))
    self.assertEqual(fun(2), cfun(2))
    self.assertEqual(fun(2), (2, 2))
    self.assertEqual(fun(3), cfun(3))
    self.assertEqual(fun(3), (6, 12))
    self.assertEqual(fun(4), cfun(4))
    self.assertEqual(fun(4), (8, 16))

  def testSwitch(self):
    def branch(x):
      y = lax.mul(2, x)
      return y, lax.mul(2, y)

    branches = [lambda x: (x, x),
                branch,
                lambda x: (x, -x)]

    def fun(x):
      if x <= 0:
        return branches[0](x)
      elif x == 1:
        return branches[1](x)
      else:
        return branches[2](x)

    def cfun(x):
      return lax.switch(x, branches, x)

    self.assertEqual(fun(-1), cfun(-1))
    self.assertEqual(fun(0), cfun(0))
    self.assertEqual(fun(1), cfun(1))
    self.assertEqual(fun(2), cfun(2))
    self.assertEqual(fun(3), cfun(3))

    cfun = api.jit(cfun)

    self.assertEqual(fun(-1), cfun(-1))
    self.assertEqual(fun(0), cfun(0))
    self.assertEqual(fun(1), cfun(1))
    self.assertEqual(fun(2), cfun(2))
    self.assertEqual(fun(3), cfun(3))

  def testSwitchResidualsMerge(self):
    def get_conds(fun):
      jaxpr = api.make_jaxpr(api.grad(fun))(0., 0)
      return [eqn for eqn in jaxpr.jaxpr.eqns if eqn.primitive.name == 'cond']

    def branch_invars_len(cond_eqn):
      lens = [len(jaxpr.jaxpr.invars) for jaxpr in cond_eqn.params['branches']]
      assert len(set(lens)) == 1
      return lens[0]

    def branch_outvars_len(cond_eqn):
      lens = [len(jaxpr.jaxpr.outvars) for jaxpr in cond_eqn.params['branches']]
      assert len(set(lens)) == 1
      return lens[0]

    branches1 = [
        lambda x: jnp.sin(x),
        lambda x: jnp.cos(x)]   # branch residuals overlap, should be reused
    branches2 = branches1 + [
        lambda x: jnp.sinh(x)]  # another overlapping residual, expect reuse
    branches3 = branches2 + [
        lambda x: jnp.sin(x) + jnp.cos(x)]  # requires one more residual slot
    def fun1(x, i):
      return lax.switch(i + 1, branches1, x)
    def fun2(x, i):
      return lax.switch(i + 1, branches2, x)
    def fun3(x, i):
      return lax.switch(i + 1, branches3, x)

    fwd1, bwd1 = get_conds(fun1)
    fwd2, bwd2 = get_conds(fun2)
    fwd3, bwd3 = get_conds(fun3)

    fwd1_num_out = branch_outvars_len(fwd1)
    fwd2_num_out = branch_outvars_len(fwd2)
    fwd3_num_out = branch_outvars_len(fwd3)
    assert fwd1_num_out == fwd2_num_out
    assert fwd3_num_out == fwd2_num_out + 1

    bwd1_num_in = branch_invars_len(bwd1)
    bwd2_num_in = branch_invars_len(bwd2)
    bwd3_num_in = branch_invars_len(bwd3)
    assert bwd1_num_in == bwd2_num_in
    assert bwd3_num_in == bwd2_num_in + 1

  def testOneBranchSwitch(self):
    branch = lambda x: -x
    f = lambda i, x: lax.switch(i, [branch], x)
    x = 7.
    self.assertEqual(f(-1, x), branch(x))
    self.assertEqual(f(0, x), branch(x))
    self.assertEqual(f(1, x), branch(x))
    cf = api.jit(f)
    self.assertEqual(cf(-1, x), branch(x))
    self.assertEqual(cf(0, x), branch(x))
    self.assertEqual(cf(1, x), branch(x))
    cf = api.jit(f, static_argnums=0)
    self.assertEqual(cf(-1, x), branch(x))
    self.assertEqual(cf(0, x), branch(x))
    self.assertEqual(cf(1, x), branch(x))

  def testIssue1379(self):

    def fun(pred):
      return lax.cond(pred, lambda x: (True, x), lambda x: (False, x), pred)

    @api.jit
    def cfun(pred):
      return fun(pred)

    self.assertEqual(fun(0), cfun(0), (False,0))
    self.assertEqual(fun(0.), cfun(0.), (False,0.))
    self.assertEqual(fun(1), cfun(1), (True,1))
    self.assertEqual(fun(1.), cfun(1.), (True,1.))

    # test that proper errors are raised for wrong types
    for pred in ["abc", [], [1,2]]:
      for f in [fun, cfun]:
        self.assertRaises(TypeError, f, pred)

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testNestedCond(self, cond):
    def fun(x):
      if x < 2:
        return lax.mul(2, x)
      else:
        if x < 5:
          return lax.mul(3, x)
        else:
          return lax.mul(4, x)

    @api.jit
    def cfun(x):
      return cond(
          lax.lt(x, 2),
          lambda x: lax.mul(2, x),
          lambda x: cond(lax.lt(x, 5),
                         x, lambda x: lax.mul(3, x),
                         4, lambda y: lax.mul(y, x)),
          x)

    self.assertEqual(cfun(1), 2)
    self.assertEqual(cfun(3), 9)
    self.assertEqual(cfun(6), 24)
    self.assertEqual(cfun(1), fun(1))
    self.assertEqual(cfun(3), fun(3))
    self.assertEqual(cfun(6), fun(6))

  def testCondTypeErrors(self):
    """Test typing error messages for  cond."""
    with self.assertRaisesRegex(TypeError,
        re.escape("Pred type must be either boolean or number, got <function")):
      lax.cond(lambda x: True, lambda top: 2., lambda fop: 3., 1.)
    with self.assertRaisesRegex(TypeError,
        re.escape("Pred type must be either boolean or number, got foo.")):
      lax.cond("foo", lambda top: 2., lambda fop: 3., 1.)
    with self.assertRaisesRegex(TypeError,
        re.escape("Pred must be a scalar, got (1.0, 1.0) of shape (2,).")):
      lax.cond((1., 1.), lambda top: 2., lambda fop: 3., 1.)
    with self.assertRaisesRegex(TypeError,
        re.escape("true_fun and false_fun output must have same type structure, got * and PyTreeDef(tuple, [*,*]).")):
      lax.cond(True, lambda top: 2., lambda fop: (3., 3.), 1.)
    with self.assertRaisesRegex(
        TypeError, textwrap.dedent(
            r"""
            true_fun and false_fun output must have identical types, got
            ShapedArray\(float32\[1\]\)
            and
            ShapedArray\(float32\[\].*\).""").strip()):
      lax.cond(True,
               lambda top: jnp.array([1.], jnp.float32),
               lambda fop: jnp.float32(1.),
               1.)

  def testSwitchErrors(self):
    """Test typing error messages for switch."""
    with self.assertRaisesRegex(TypeError,
        re.escape("Index type must be an integer, got <function")):
      lax.switch(lambda x: True, [lambda _: 2., lambda _: 3.], 1.)
    with self.assertRaisesRegex(TypeError,
        re.escape("Index type must be an integer, got foo.")):
      lax.switch("foo", [lambda _: 2., lambda _: 3.], 1.)
    with self.assertRaisesRegex(TypeError,
        re.escape("Branch index must be scalar, got (1.0, 1.0) of shape (2,).")):
      lax.switch((1., 1.), [lambda _: 2., lambda _: 3.], 1.)
    with self.assertRaisesRegex(ValueError,
        re.escape("Empty branch sequence")):
      lax.switch(0, [], 1.)
    with self.assertRaisesRegex(TypeError,
        re.escape("branch 0 and 1 outputs must have same type structure, got * and PyTreeDef(tuple, [*,*]).")):
      lax.switch(1, [lambda _: 2., lambda _: (3., 3.)], 1.)
    with self.assertRaisesRegex(
        TypeError, textwrap.dedent(
            r"""
            branch 0 and 1 outputs must have identical types, got
            ShapedArray\(float32\[1\]\)
            and
            ShapedArray\(float32\[\].*\).""").strip()):
      lax.switch(1, [lambda _: jnp.array([1.], jnp.float32),
                     lambda _: jnp.float32(1.)],
                 1.)

  def testCondOneBranchConstant(self):
    def fun(x):
      if x < 3:
        return 5.
      else:
        return x

    @api.jit
    def cfun(x):
      return lax.cond(lax.lt(x, 3), lambda x: 5, lambda x: x, x)

    self.assertEqual(fun(0), cfun(0))
    self.assertEqual(cfun(0), 5)
    self.assertEqual(fun(4), cfun(4))
    self.assertEqual(cfun(4), 4)

  def testCondOneBranchConstantTuple(self):
    def fun(x):
      if x < 3:
        return (1., 2., 3.)
      else:
        return (x, 2., 4.)

    @api.jit
    def cfun(x):
      return lax.cond(lax.lt(x, 3),
                      lambda x: (1, 2., 3.),
                      lambda x: (x, 2., 4.),
                      x)

    self.assertEqual(fun(0), cfun(0))
    self.assertEqual(cfun(0), (1, 2., 3.))
    self.assertEqual(fun(4), cfun(4))
    self.assertEqual(cfun(4), (4, 2., 4.))

  def testCondBatched(self):
    def fun(x, y, z):
      pred = lax.lt(x, 3)
      true_fun = lambda y: y
      false_fun = lambda z: lax.neg(z)
      return lax.cond(pred, y, true_fun, z, false_fun)

    # these cases stay as cond
    x = jnp.array(2)
    y = jnp.array([1, 2])
    z = jnp.array([3, 4])
    ans = api.vmap(fun, (None, 0, 0))(x, y, z)
    jaxpr = api.make_jaxpr(api.vmap(fun, (None, 0, 0)))(x, y, z)
    expected = np.array([1, 2])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" not in str(jaxpr)

    x = jnp.array(4)
    ans = api.vmap(fun, (None, 0, 0))(x, y, z)
    jaxpr = api.make_jaxpr(api.vmap(fun, (None, 0, 0)))(x, y, z)
    expected = np.array([-3, -4])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" not in str(jaxpr)

    fun = api.jit(fun)
    ans = api.vmap(fun, (None, 0, 0))(x, y, z)
    expected = np.array([-3, -4])
    self.assertAllClose(ans, expected, check_dtypes=False)

    z = jnp.array(5)
    ans = api.vmap(fun, (None, 0, None))(x, y, z)
    jaxpr = api.make_jaxpr(api.vmap(fun, (None, 0, None)))(x, y, z)
    expected = np.array([-5, -5])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" not in str(jaxpr)


    # these cases become select
    x = jnp.array([2, 4])
    ans = api.vmap(fun, (0, 0, None))(x, y, z)
    jaxpr = api.make_jaxpr(api.vmap(fun, (0, 0, None)))(x, y, z)
    expected = np.array([1, -5])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" in str(jaxpr)

    z = jnp.array([3, 4])
    ans = api.vmap(fun)(x, y, z)
    jaxpr = api.make_jaxpr(api.vmap(fun))(x, y, z)
    expected = np.array([1, -4])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" in str(jaxpr)

  def testSwitchBatched(self):
    def fun(index, x, y, z):
      branches = [lambda xyz: xyz[0],
                  lambda xyz: lax.neg(xyz[1]),
                  lambda xyz: lax.sign(xyz[2])]
      return lax.switch(index, branches, (x, y, z))

    # these cases stay as cond
    x = jnp.array(0)
    y = jnp.array([1, 2])
    z = jnp.array([3, 4])
    w = jnp.array(9)
    ans = api.vmap(fun, (None, 0, 0, None))(x, y, z, w)
    jaxpr = api.make_jaxpr(api.vmap(fun, (None, 0, 0, None)))(x, y, z, w)
    expected = np.array([1, 2])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" not in str(jaxpr)

    x = jnp.array(1)
    ans = api.vmap(fun, (None, 0, 0, None))(x, y, z, w)
    jaxpr = api.make_jaxpr(api.vmap(fun, (None, 0, 0, None)))(x, y, z, w)
    expected = np.array([-3, -4])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" not in str(jaxpr)

    fun = api.jit(fun)
    ans = api.vmap(fun, (None, 0, 0, None))(x, y, z, w)
    expected = np.array([-3, -4])
    self.assertAllClose(ans, expected, check_dtypes=False)

    z = jnp.array(5)
    ans = api.vmap(fun, (None, 0, None, None))(x, y, z, w)
    jaxpr = api.make_jaxpr(api.vmap(fun, (None, 0, None, None)))(x, y, z, w)
    expected = np.array([-5, -5])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" not in str(jaxpr)


    # these cases become select
    x = jnp.array([0, 1])
    ans = api.vmap(fun, (0, 0, None, None))(x, y, z, w)
    jaxpr = api.make_jaxpr(api.vmap(fun, (0, 0, None, None)))(x, y, z, w)
    expected = np.array([1, -5])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" in str(jaxpr)

    z = jnp.array([3, 4])
    w = jnp.array([9, 9])
    ans = api.vmap(fun)(x, y, z, w)
    jaxpr = api.make_jaxpr(api.vmap(fun))(x, y, z, w)
    expected = np.array([1, -4])
    self.assertAllClose(ans, expected, check_dtypes=False)
    assert "select" in str(jaxpr)

  def testCondJVP(self):
    def fun_ref(x):
      if x < 3:
        return (x, x)
      else:
        y = 2 * x
        return y, 2 * y

    def fun(x):
      def false_fun(x):
        y = 2 * x
        return y, 2 * y
      return lax.cond(x < 3, lambda x: (x, x), false_fun, x)

    x = 3.14
    ans = api.jvp(fun, (x,), (x,))
    expected = api.jvp(fun_ref, (x,), (x,))
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x,), order=2, modes=["fwd"])

    x = 2.72
    ans = api.jvp(fun, (x,), (x,))
    expected = api.jvp(fun_ref, (x,), (x,))
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x,), order=2, modes=["fwd"])

  def testSwitchJVP(self):
    def branch(x):
      y = 2 * x
      return y, 2 * y

    branches = [lambda x: (x, x),
                branch,
                lambda x: (x, -x)]

    def fun_ref(x):
      idx = x // 1
      if idx <= 0:
        return branches[0](x)
      elif idx == 1:
        return branches[1](x)
      else:
        return branches[2](x)

    def fun(x):
      idx = lax.convert_element_type(x // 1, np.int32)
      return lax.switch(idx, branches, x)

    for x in [-0.7, 0.7, 1.7, 2.7, 3.7]:
      ans = api.jvp(fun, (x,), (x,))
      expected = api.jvp(fun_ref, (x,), (x,))
      self.assertAllClose(ans, expected, check_dtypes=False)
      jtu.check_grads(fun, (x,), order=2, modes=["fwd"])

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondJVP2(self, cond):
    def fun_ref(x):
      if x < 3:
        return 2.
      else:
        return 2. * x

    def fun(x):
      return cond(x < 3, (), lambda _: 2., x, lambda x: 2. * x)

    x = 3.14
    ans = api.jvp(fun, (x,), (x,))
    expected = api.jvp(fun_ref, (x,), (x,))
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x,), order=2, modes=["fwd"])

    x = 2.72
    ans = api.jvp(fun, (x,), (x,))
    expected = api.jvp(fun_ref, (x,), (x,))
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x,), order=2, modes=["fwd"])

  def testCondGrad(self):
    def f_ref(x):
      return 3. * x if x < 2 else jnp.sin(x)

    def f(x):
      return lax.cond(x < 2, lambda x: 3. * x, lambda x: jnp.sin(x), x)

    x = 2.14
    ans = api.grad(f)(x)
    expected = api.grad(f_ref)(x)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(f, (x,), order=2, modes=["fwd", "rev"])

    x = 1.72
    ans = api.grad(f)(x)
    expected = api.grad(f_ref)(x)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(f, (x,), order=2, modes=["fwd", "rev"])

  def testSwitchGrad(self):
    branches = [lambda x: 3. * x,
                lambda x: jnp.sin(x),
                lambda x: -x]

    def f_ref(x):
      idx = x // 1
      if idx <= 0:
        return branches[0](x)
      elif idx == 1:
        return branches[1](x)
      else:
        return branches[2](x)

    def f(x):
      idx = lax.convert_element_type(x // 1, np.int32)
      return lax.switch(idx, branches, x)

    for x in [-0.7, 0.7, 1.7, 2.7, 3.7]:
      ans = api.grad(f)(x)
      expected = api.grad(f_ref)(x)
      self.assertAllClose(ans, expected, check_dtypes=False)
      jtu.check_grads(f, (x,), order=2, modes=["fwd", "rev"])

  def testSwitchGradWithWeakTypeMismatch(self):  # issue #4696, PR #4896
    dtype = jnp.ones(1).dtype
    dtype = jnp.float32 if dtype == jnp.float32 else jnp.float64

    branches = [
        lambda x: x,             # This preserves the weak type of x.
        lambda x: x + dtype(1),  # This strips the weak type of x.
    ]

    def f_ref(x):
      i = x.astype(jnp.int32)
      return branches[i](x)

    def f(x):
      return lax.switch(x.astype(jnp.int32), branches, x)

    for x in [0., 1.]:
      ans = api.grad(f)(x)
      expected = api.grad(f_ref)(x)
      self.assertAllClose(ans, expected, check_dtypes=False)

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondGrad2(self, cond):
    def f_ref(x):
      z = jnp.array([1., 2.]) * x if x[0] < 2 else jnp.sin(x)
      return z.sum()

    def _f(x):
      return cond(
          x[0] < 2,
          lambda x: jnp.array([1., 2.]) * x,
          lambda x: jnp.sin(x),
          x)

    f = lambda x: api.jit(_f)(x).sum()

    x = 2.14 * jnp.ones(2)
    ans = api.grad(f)(x)
    expected = api.grad(f_ref)(x)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(f, (x,), order=2, modes=["fwd", "rev"])

    x = 1.72 * jnp.ones(2)
    ans = api.grad(f)(x)
    expected = api.grad(f_ref)(x)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(f, (x,), order=2, modes=["fwd", "rev"],
                    rtol={jnp.float32: 1e-2, jnp.float64: 2e-3})

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondGrad3(self, cond):
    def fun_ref(x):
      if x < 3:
        return 2.
      else:
        return 2. * x

    def fun(x):
      return cond(x < 3, (), lambda _: 2., x, lambda x: 2. * x)

    x = 3.14
    ans = api.grad(fun)(x)
    expected = api.grad(fun_ref)(x)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x,), order=2, modes=["fwd", "rev"])

    x = 2.72
    ans = api.grad(fun)(x)
    expected = api.grad(fun_ref)(x)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x,), order=2, modes=["fwd", "rev"])

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondGrad4(self, cond):
    def fun_ref(x, y):
      if x < 3:
        return 2. * jnp.sin(y)
      else:
        return 2. * jnp.cos(x)

    def fun(x, y):
      return cond(
          x < 3,
          (), lambda _: 2. * jnp.sin(y),
          x,  lambda x: 2. * x)

    y = 5.8
    x = 3.14
    ans = api.grad(fun, 1)(x, y)
    expected = api.grad(fun_ref, 1)(x, y)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x, y), order=2, modes=["fwd", "rev"])

    x = 2.72
    ans = api.grad(fun, 1)(x, y)
    expected = api.grad(fun_ref, 1)(x, y)
    self.assertAllClose(ans, expected, check_dtypes=False)
    jtu.check_grads(fun, (x, y), order=2, modes=["fwd", "rev"])

  def testCondLinearize(self):
    def f(x):
      return lax.cond(x < 2, lambda x: 3. * x, lambda x: jnp.sin(x), x)
    y, f_lin = api.linearize(f, 1.)
    self.assertAllClose(y, 3., check_dtypes=False)
    self.assertAllClose(f_lin(2.), 6., check_dtypes=False)
    y, f_lin = api.linearize(f, 4.)
    self.assertAllClose(y, jnp.sin(4.), check_dtypes=False)
    self.assertAllClose(f_lin(2.), jnp.cos(4.) * 2., check_dtypes=False)

  def testSwitchLinearize(self):
    branches = [lambda x: 3. * x,
                lambda x: jnp.sin(x),
                lambda x: -x]
    def f(x):
      idx = lax.convert_element_type(x // 1, np.int32)
      return lax.switch(idx, branches, x)

    # branch 0
    y, f_lin = api.linearize(f, -1.)
    self.assertAllClose(y, -3., check_dtypes=False)
    self.assertAllClose(f_lin(2.), 6., check_dtypes=False)
    y, f_lin = api.linearize(f, 0.)
    self.assertAllClose(y, 0., check_dtypes=False)
    self.assertAllClose(f_lin(2.), 6., check_dtypes=False)

    # branch 1
    y, f_lin = api.linearize(f, 1.)
    self.assertAllClose(y, jnp.sin(1.), check_dtypes=False)
    self.assertAllClose(f_lin(2.), jnp.cos(1.) * 2., check_dtypes=False)

    # branch 2
    y, f_lin = api.linearize(f, 2.)
    self.assertAllClose(y, -2., check_dtypes=False)
    self.assertAllClose(f_lin(2.), -2., check_dtypes=False)
    y, f_lin = api.linearize(f, 3.)
    self.assertAllClose(y, -3., check_dtypes=False)
    self.assertAllClose(f_lin(2.), -2., check_dtypes=False)

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondLinearize2(self, cond):
    def f_ref(x):
      z = jnp.array([1., 2.]) * x if x[0] < 2 else jnp.cos(jnp.sin(x))
      return z.sum()

    def f(x):
      return cond(
          x[0] < 2,
          lambda x: jnp.array([1., 2.]) * x,
          lambda x: jnp.cos(jnp.sin(x)),
          x).sum()

    x = 2.14 * jnp.ones(2)
    y, f_lin = api.linearize(f, x)
    y_ref, f_lin_ref = api.linearize(f_ref, x)
    self.assertAllClose(y, y_ref, check_dtypes=False)
    self.assertAllClose(f_lin(x), f_lin_ref(x), check_dtypes=False)

    x = -2.14 * jnp.ones(2)
    y, f_lin = api.linearize(f, x)
    y_ref, f_lin_ref = api.linearize(f_ref, x)
    self.assertAllClose(y, y_ref, check_dtypes=False)
    self.assertAllClose(f_lin(x), f_lin_ref(x), check_dtypes=False)

    f = api.jit(f)
    x = 2.14 * jnp.ones(2)
    y, f_lin = api.linearize(f, x)
    y_ref, f_lin_ref = api.linearize(f_ref, x)
    self.assertAllClose(y, y_ref, check_dtypes=False)
    self.assertAllClose(f_lin(x), f_lin_ref(x), check_dtypes=False)

  def testCondJit(self):
    def f(x):
      return lax.cond(x < 2, lambda x: 3. * x, lambda x: jnp.sin(x), x)
    y = api.jit(f)(1.)
    expected = f(1.)
    self.assertAllClose(y, expected, check_dtypes=False)
    y = api.jit(f)(4.)
    expected = f(4.)
    self.assertAllClose(y, expected, check_dtypes=False)

  def testSwitchJit(self):
    branches = [lambda x: 3. * x,
                lambda x: jnp.sin(x),
                lambda x: -x]
    def f(x):
      idx = lax.convert_element_type(x // 1, np.int32)
      return lax.switch(idx, branches, x)
    for x in [-1., 0., 1., 2., 3.]:
      y = api.jit(f)(x)
      expected = f(x)
      self.assertAllClose(y, expected, check_dtypes=False)

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondJitDisabled(self, cond):
    def f_ref(x):
      return 3. * x if x < 2 else jnp.sin(x)
    def f(x):
      return cond(x < 2, lambda x: 3. * x, lambda x: jnp.sin(x), x)

    with api.disable_jit():
      y = f(1.)
      expected = f_ref(1.)
      self.assertAllClose(y, expected, check_dtypes=False)

    with api.disable_jit():
      y = api.jit(f)(1.)
      expected = f(1.)
      self.assertAllClose(y, expected, check_dtypes=False)

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondWithConsts(self, cond):
    def f(x):
      return cond(x < 2,
                  lambda x: np.array([1., 2.]) * x,
                  lambda x: np.array([3., 4.]) * jnp.sin(x),
                  x)

    def f_ref(x):
      if x < 2:
        return np.array([1., 2.]) * x
      else:
        return np.array([3., 4.]) * np.sin(x)

    y = f(1.)
    expected = f_ref(1.)
    self.assertAllClose(y, expected, check_dtypes=False)
    y = f(4.)
    expected = f_ref(4.)
    self.assertAllClose(y, expected, check_dtypes=False)

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondJitWithConsts(self, cond):
    def f(x):
      return cond(x < 2,
                  lambda x: np.array([1., 2.]) * x,
                  lambda x: np.array([3., 4.]) * jnp.sin(x),
                  x)

    y = api.jit(f)(1.)
    expected = f(1.)
    self.assertAllClose(y, expected, check_dtypes=False)
    y = api.jit(f)(4.)
    expected = f(4.)
    self.assertAllClose(y, expected, check_dtypes=False)

  @parameterized.named_parameters(
      {"testcase_name": f"_{name}", "cond": cond}
      for cond, name in COND_IMPLS)
  def testCondVmapGrad(self, cond):
    # https://github.com/google/jax/issues/2264
    def f_1(x): return x ** 2
    def f_2(x): return x ** 3

    def f(x): return cond(x > 0, f_1, f_2, x)
    def g(x): return jnp.where(x > 0, f_1(x), f_2(x))

    x = jnp.linspace(-1, 1, 20)
    ans = api.vmap(api.grad(f))(x)
    expected = api.vmap(api.grad(g))(x)
    self.assertAllClose(ans, expected, check_dtypes=False)

  def testIssue1263(self):
    def f(rng, x):
      cond = random.bernoulli(rng)
      return lax.cond(cond, x, lambda x: x, jnp.abs(x) - 1., lambda x: x)

    def body_fn(i, state):
      rng, x = state
      key, subkey = random.split(rng)
      return key, f(subkey, x)

    def g(rng, x):
      return lax.fori_loop(0, 10, body_fn, (rng, x))

    api.vmap(g)(random.split(random.PRNGKey(0), 3), jnp.ones((3, 4)))

  def testIssue514(self):
    # just check this doesn't crash
    lax.cond(True,
            (0, 0), lambda x: (x[0], 0),
            (1, 1), lambda x: x)

  def testIssue649(self):
    from jax import lax

    def body(x):
      a, b = x
      return (7, b + 1)

    def cond(x):
      a, b = x
      return b < 10

    out = lax.while_loop(cond, body, (33, 4))
    self.assertEqual(out, (7, 10))

  @parameterized.named_parameters(
      {"testcase_name": "_jit_scan={}_jit_f={}_impl={}".format(
          jit_scan, jit_f, scan_name),
       "jit_scan": jit_scan, "jit_f": jit_f, "scan": scan_impl}
      for jit_scan in [False, True]
      for jit_f in [False, True]
      for scan_impl, scan_name in SCAN_IMPLS)
  def testScanImpl(self, jit_scan, jit_f, scan):
    rng = np.random.RandomState(0)

    d = rng.randn(2)
    def f(c, a):
      assert a.shape == (3,)
      assert c.shape == (4,)
      b = jnp.cos(jnp.sum(jnp.sin(a)) + jnp.sum(jnp.cos(c)) + jnp.sum(jnp.tan(d)))
      c = jnp.sin(c * b)
      assert b.shape == ()
      return c, b

    if jit_f:
      f = api.jit(f)
    if jit_scan:
      scan = api.jit(scan, (0,))

    as_ = rng.randn(5, 3)
    c = rng.randn(4)

    ans =                scan(f, c, as_)
    expected = scan_reference(f, c, as_)
    self.assertAllClose(ans, expected, check_dtypes=False)

  @parameterized.named_parameters(
      {"testcase_name": "_jit_scan={}_jit_f={}_impl={}".format(
          jit_scan, jit_f, scan_name),
       "jit_scan": jit_scan, "jit_f": jit_f, "scan": scan_impl}
      for jit_scan in [False, True]
      for jit_f in [False, True]
      for scan_impl, scan_name in SCAN_IMPLS)
  def testScanJVP(self, jit_scan, jit_f, scan):
    rng = np.random.RandomState(0)

    d = rng.randn(2)
    def f(c, a):
      assert a.shape == (3,)
      assert c.shape == (4,)
      b = jnp.cos(jnp.sum(jnp.sin(a)) + jnp.sum(jnp.cos(c)) + jnp.sum(jnp.tan(d)))
      c = jnp.sin(c * b)
      assert b.shape == ()
      return c, b

    if jit_f:
      f = api.jit(f)
    if jit_scan:
      scan = api.jit(scan, (0,))

    as_ = rng.randn(5, 3)
    c = rng.randn(4)

    ans = api.jvp(     lambda c, as_:           scan(f, c, as_), (c, as_), (c, as_))
    expected = api.jvp(lambda c, as_: scan_reference(f, c, as_), (c, as_), (c, as_))
    self.assertAllClose(ans, expected, check_dtypes=False,
                        rtol={np.float64: 1e-14, np.float32: 1e-5})

    jtu.check_grads(partial(scan, f), (c, as_), order=2, modes=["fwd"])

  @parameterized.named_parameters(
      {"testcase_name": "_jit_scan={}_jit_f={}_impl={}".format(
          jit_scan, jit_f, scan_name),
       "jit_scan": jit_scan, "jit_f": jit_f, "scan": scan_impl}
      for jit_scan in [False, True]
      for jit_f in [False, True]
      for scan_impl, scan_name in SCAN_IMPLS)
  def testScanLinearize(self, jit_scan, jit_f, scan):
    rng = np.random.RandomState(0)

    d = rng.randn(2)
    def f(c, a):
      assert a.shape == (3,)
      assert c.shape == (4,)
      b = jnp.cos(jnp.sum(jnp.sin(a)) + jnp.sum(jnp.cos(c)) + jnp.sum(jnp.tan(d)))
      c = jnp.sin(c * b)
      assert b.shape == ()
      return c, b

    if jit_f:
      f = api.jit(f)
    if jit_scan:
      scan = api.jit(scan, (0,))

    as_ = rng.randn(5, 3)
    c = rng.randn(4)

    ans = api.linearize(lambda c, as_:                scan(f, c, as_), c, as_)[1](c, as_)
    expected = api.linearize(lambda c, as_: scan_reference(f, c, as_), c, as_)[1](c, as_)
    self.assertAllClose(ans, expected, check_dtypes=False,
                        rtol={np.float64: 1e-14})

  @parameterized.named_parameters(
      {"testcase_name": "_jit_scan={}_jit_f={}_impl={}".format(
          jit_scan, jit_f, scan_name),
       "jit_scan": jit_scan, "jit_f": jit_f, "scan": scan_impl}
      for jit_scan in [False, True]
      for jit_f in [False, True]
      for scan_impl, scan_name in SCAN_IMPLS)
  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def testScanGrad(self, jit_scan, jit_f, scan):
    rng = np.random.RandomState(0)

    d = rng.randn(2)
    def f(c, a):
      assert a.shape == (3,)
      assert c.shape == (4,)
      b = jnp.sum(jnp.sin(a)) + jnp.sum(jnp.sin(c)) + jnp.sum(jnp.sin(d))
      c = jnp.sin(c * b)
      assert b.shape == ()
      return c, b

    if jit_f:
      f = api.jit(f)
    if jit_scan:
      scan = api.jit(scan, static_argnums=(0,))

    as_ = rng.randn(5, 3)
    c = rng.randn(4)

    ans = api.grad(lambda c, as_:      list(          scan(f, c, as_))[0].sum())(c, as_)
    expected = api.grad(lambda c, as_: list(scan_reference(f, c, as_))[0].sum())(c, as_)
    self.assertAllClose(ans, expected, check_dtypes=False,
                        rtol={np.float32: 2e-5, np.float64: 1e-13})

    jtu.check_grads(partial(scan, f), (c, as_), order=2, modes=["rev"],
                    atol=1e-3, rtol=5e-3)

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def testScanRnn(self):
    r = npr.RandomState(0)

    n_in = 4
    n_hid = 2
    n_out = 1
    length = 3

    W_trans = r.randn(n_hid, n_hid + n_in).astype(jnp.float_)
    W_out = r.randn(n_out, n_hid + n_in).astype(jnp.float_)
    params = W_trans, W_out

    inputs = r.randn(length, n_in).astype(jnp.float_)
    targets = r.randn(length, n_out).astype(jnp.float_)

    def step(params, state, input):
      W_trans, W_out = params
      stacked = jnp.concatenate([state, input])
      output = jnp.tanh(jnp.dot(W_out, stacked))
      next_state = jnp.tanh(jnp.dot(W_trans, stacked))
      return next_state, output

    def rnn(params, inputs):
      init_state = jnp.zeros(n_hid)
      _, outputs = lax.scan(partial(step, params), init_state, inputs)
      return outputs

    def loss(params, inputs, targets):
      predictions = rnn(params, inputs)
      return jnp.sum((predictions - targets)**2)

    # evaluation doesn't crash
    loss(params, inputs, targets)

    # jvp evaluation doesn't crash
    api.jvp(lambda params: loss(params, inputs, targets), (params,), (params,))

    # jvp numerical check passes
    jtu.check_grads(loss, (params, inputs, targets), order=2, modes=["fwd"],
                    rtol={np.float32: 2e-2, np.float64: 1e-6})

    # linearize works
    _, expected = api.jvp(loss, (params, inputs, targets),
                          (params, inputs, targets))
    _, linfun = api.linearize(loss, params, inputs, targets)
    ans = linfun(params, inputs, targets)
    self.assertAllClose(ans, expected, check_dtypes=False)

    # gradient evaluation doesn't crash
    api.grad(loss)(params, inputs, targets)

    # gradient check passes
    jtu.check_grads(loss, (params, inputs, targets), order=2, rtol=2e-2)

    # we can vmap to batch things
    batch_size = 7
    batched_inputs = r.randn(batch_size, length, n_in).astype(jnp.float_)
    batched_targets = r.randn(batch_size, length, n_out).astype(jnp.float_)
    batched_loss = api.vmap(lambda x, y: loss(params, x, y))
    losses = batched_loss(batched_inputs, batched_targets)
    expected = np.stack(list(map(lambda x, y: loss(params, x, y),
                                  batched_inputs, batched_targets)))
    self.assertAllClose(losses, expected, check_dtypes=False, rtol=1e-2)

  def testIssue711(self):
    # Tests reverse-mode differentiation through a scan for which the scanned
    # function also involves reverse-mode differentiation.
    # See https://github.com/google/jax/issues/711
    def harmonic_bond(conf, params):
      return jnp.sum(conf * params)

    def minimize_structure(test_params):
      energy_fn = partial(harmonic_bond, params=test_params)

      def apply_carry(carry, _):
        i, x = carry
        new_x = x - 0.1 * api.grad(energy_fn)(x)
        new_carry = (i+1, new_x)
        return new_carry, _

      x0 = jnp.array([1., 2., 3.])
      carry_final, _ = lax.scan(apply_carry, (0, x0), jnp.zeros((75, 0)))
      _, x_final = carry_final
      return x_final

    initial_params = 0.5
    minimize_structure(initial_params)  # doesn't crash

    def loss(test_params):
      x_final = minimize_structure(test_params)
      return jnp.sum(jnp.sin(1.0 - x_final))

    api.grad(loss)(0.25)  # doesn't crash

  def testIssue744(self):
    Point = collections.namedtuple('Point', ['x', 'y'])
    p0 = Point(x=jnp.array(1), y=jnp.array(2))

    def plus_one(p, iter_idx):
      return Point(p.x+1, p.y+1), iter_idx

    self.assertRaisesRegex(
        ValueError,
        'scan got value with no leading axis to scan over.*',
        lambda: lax.scan(plus_one, p0, list(range(5))))

  def testScanTypeErrors(self):
    """Test typing error messages for scan."""
    a = jnp.arange(5)
    # Body output not a tuple
    with self.assertRaisesRegex(TypeError,
        re.escape("scan body output must be a pair, got ShapedArray(float32[]).")):
      lax.scan(lambda c, x: np.float32(0.), 0, a)
    with  self.assertRaisesRegex(TypeError,
        re.escape("scan carry output and input must have same type structure, "
                  "got PyTreeDef(tuple, [*,*,*]) and PyTreeDef(tuple, [*,PyTreeDef(tuple, [*,*])])")):
      lax.scan(lambda c, x: ((0, 0, 0), x), (1, (2, 3)), a)
    with self.assertRaisesRegex(TypeError,
        re.escape("scan carry output and input must have same type structure, got * and PyTreeDef(None, []).")):
      lax.scan(lambda c, x: (0, x), None, a)
    with self.assertRaisesWithLiteralMatch(
        TypeError,
        "scan carry output and input must have identical types, got\n"
        "ShapedArray(int32[])\n"
        "and\n"
        "ShapedArray(float32[])."):
      lax.scan(lambda c, x: (np.int32(0), x), np.float32(1.0), a)
    with self.assertRaisesRegex(TypeError,
        re.escape("scan carry output and input must have same type structure, got * and PyTreeDef(tuple, [*,*]).")):
      lax.scan(lambda c, x: (0, x), (1, 2), jnp.arange(5))


  @parameterized.named_parameters(
      {"testcase_name": "_{}".format(scan_name),
       "scan": scan_impl}
      for scan_impl, scan_name in SCAN_IMPLS)
  def testScanHigherOrderDifferentiation(self, scan):
    d = 0.75
    def f(c, a):
      b = jnp.sin(c * jnp.sum(jnp.cos(d * a)))
      c = 0.9 * jnp.cos(d * jnp.sum(jnp.sin(c * a)))
      return c, b

    as_ = jnp.arange(6.).reshape((3, 2))
    c = 1.

    jtu.check_grads(lambda c, as_: scan(f, c, as_), (c, as_),
                    modes=["rev"], order=2, rtol={np.float32: 6e-3})

  @parameterized.named_parameters(
      {"testcase_name": "_jit_scan={}_jit_f={}_in_axes={}_impl={}".format(
          jit_scan, jit_f, in_axes, scan_name),
       "jit_scan": jit_scan, "jit_f": jit_f, "in_axes": in_axes,
       "scan": scan_impl}
      for jit_scan in [False, True]
      for jit_f in [False, True]
      for scan_impl, scan_name in SCAN_IMPLS
      for in_axes in itertools.product([None, 0, 1], [None, 0, 1, 2])
      if in_axes != (None, None))
  def testScanVmap(self, jit_scan, jit_f, in_axes, scan):
    rng = np.random.RandomState(0)

    d = rng.randn(2)
    def f(c, a):
      assert a.shape == (3,)
      assert c.shape == (4,)
      b = jnp.cos(jnp.sum(jnp.sin(a)) + jnp.sum(jnp.cos(c)) + jnp.sum(jnp.tan(d)))
      c = jnp.sin(c * b)
      assert b.shape == ()
      return c, b

    if jit_f:
      f = api.jit(f)
    if jit_scan:
      scan = api.jit(scan, (0,))

    as_shape = [5, 3]
    c_shape = [4]

    c_bdim, as_bdim = in_axes
    if c_bdim is not None:
      c_shape.insert(c_bdim, 7)
    if as_bdim is not None:
      as_shape.insert(as_bdim, 7)

    as_ = rng.randn(*as_shape)
    c = rng.randn(*c_shape)

    ans = api.vmap(lambda c, as_:                scan(f, c, as_), in_axes)(c, as_)
    expected = api.vmap(lambda c, as_: scan_reference(f, c, as_), in_axes)(c, as_)
    self.assertAllClose(ans, expected, check_dtypes=False)

  def testScanVmapTuples(self):
    def f(c, a):
      a1, a2 = a
      c1, c2 = c
      b = jnp.sum(jnp.cos(a1)) * jnp.sum(jnp.tan(c2 * a2))
      c = c1 * jnp.sin(jnp.sum(a1 * a2)), c2 * jnp.cos(jnp.sum(a1))
      return c, b

    in_axes = (0, (1, 2))

    r = np.random.RandomState(0)
    as_ = (r.randn(3, 7), r.randn(3, 4, 7))
    c = (r.randn(7, 2), r.randn(7))

    expected_c_out, expected_bs = [], []
    for i in range(7):
      c_out, bs = lax.scan(f, (c[0][i], c[1][i]), (as_[0][:,i], as_[1][:,:,i]))
      expected_c_out.append(c_out)
      expected_bs.append(bs)
    expected_c_out_0, expected_c_out_1 = unzip2(expected_c_out)
    expected_c_out = (jnp.stack(expected_c_out_0), jnp.stack(expected_c_out_1))
    expected_bs = jnp.stack(expected_bs)
    expected = expected_c_out, expected_bs

    ans = api.vmap(lambda c, as_:            lax.scan(f, c, as_), in_axes)(c, as_)
    self.assertAllClose(ans, expected, check_dtypes=False)

  def testScanVmapFixpoint(self):
    def f(carry_init):
      def scan_body(c, x):
        # The carry is a 4-tuple, the last element starts batched,
        # and the carry is shifted left at each iteration.
        return ((c[1], c[2], c[3], 0.), None)
      return lax.scan(scan_body, (0., 1., 2., carry_init), jnp.zeros(2))
    carry_init = jnp.array([3., 4., 5.])
    carry_out, _ = api.vmap(f)(carry_init)
    self.assertAllClose(carry_out[3], jnp.array([0., 0., 0.]), check_dtypes=False)
    self.assertAllClose(carry_out[2], jnp.array([0., 0., 0.]), check_dtypes = False)
    # After two shifts, we get the carry_init
    self.assertAllClose(carry_out[1], carry_init, check_dtypes=False)
    self.assertAllClose(carry_out[0], jnp.array([2., 2., 2.]), check_dtypes = False)

  # TODO(mattjj, dougalm): fix this test when skip_checks is False
  def testIssue757(self):
    # code from https://github.com/google/jax/issues/757
    def fn(a):
        return jnp.cos(a)

    def loop(val):
        iterations = 10
        def apply_carry(x, i):
            return api.grad(fn, argnums=(0,))(x)[0], i

        final_val, _ = lax.scan(
            apply_carry,
            val,
            jnp.arange(iterations)
        )
        return final_val

    arg = 0.5
    api.jit(api.jacfwd(loop, argnums=(0,)))(arg)  # doesn't crash

  def testIssue804(self):
    num_devices = xla_bridge.device_count()
    f = partial(lax.scan, lambda c, x: (c + lax.psum(x, "i") , c), 0.)
    api.pmap(f, axis_name="i")(jnp.ones((num_devices, 4)))  # doesn't crash

  def testMap(self):
    f = lambda x: x ** 2
    xs = jnp.arange(10)
    expected = xs ** 2
    actual = lax.map(f, xs)
    self.assertAllClose(actual, expected)

  def testMapEmpty(self):
    # https://github.com/google/jax/issues/2412
    ans = lax.map(lambda x: x * x, jnp.array([]))
    expected = jnp.array([])
    self.assertAllClose(ans, expected)

  def testCaching(self):
    def cond(x):
      assert python_should_be_executing
      return x < 5

    def body(x):
      assert python_should_be_executing
      return x + 2

    python_should_be_executing = True
    lax.while_loop(cond, body, 0)

    python_should_be_executing = False
    lax.while_loop(cond, body, 0)

  def testCaching2(self):
    # This second caching test shows a different kind of caching that we haven't
    # implemented (but could!), namely that Python functions that are distinct
    # objects but are equivalent functions trigger cache hits. This kind of
    # caching could be salient when using lambda functions with control flow:
    #
    #   lax.while_loop(lambda x: x < 5, lambda x: x + 2, 0)
    #   lax.while_loop(lambda x: x < 5, lambda x: x + 2, 0)
    #
    # To get a cache hit on the second line we'd need to form a jaxpr and
    # compare them for equality (including the literals on identity). We could
    # implement that by adding a __hash__/__eq__ to core.Jaxpr and
    # core.ClosedJaxpr (see #1221).
    raise SkipTest("not implemented")

    def cond(x):
      assert python_should_be_executing
      return x < 5

    def body(x):
      assert python_should_be_executing
      return x + 2

    python_should_be_executing = True
    lax.while_loop(cond, body, 0)

    def cond(x):
      assert python_should_be_executing
      return x < 5

    def body(x):
      assert python_should_be_executing
      return x + 2

    python_should_be_executing = False
    lax.while_loop(cond, body, 0)

  def testWhileCondConstant(self):
    out = lax.while_loop(lambda _: False, lambda _: (), ())  # doesn't crash
    self.assertEqual(out, ())

  @parameterized.named_parameters(
      {"testcase_name": "_jit_loop={}_jit_body={}_jit_cond={}".format(
          jit_loop, jit_body, jit_cond),
       "jit_loop": jit_loop, "jit_body": jit_body, "jit_cond": jit_cond}
      for jit_loop in [False, True]
      for jit_body in [False, True]
      for jit_cond in [False, True])
  def testWhileJVP(self, jit_loop=True, jit_body=False, jit_cond=True):
    cond = lambda x: x[0, 2] <= 8
    body = lambda x: x * x

    if jit_cond:
      cond = api.jit(cond)
    if jit_body:
      body = api.jit(body)

    loop = partial(lax.while_loop, cond, body)
    if jit_loop:
      loop = api.jit(loop)

    loop_ref = partial(while_loop_reference, cond, body)

    x = jnp.arange(9.).reshape((3, 3))
    ans = api.jvp(loop, (x,), (x,))
    expected = api.jvp(loop_ref, (x,), (x,))
    self.assertAllClose(ans, expected, check_dtypes=False)

    jtu.check_grads(loop, (x,), order=2, modes=["fwd"])

  def testWhileJVPViaForiLoop(self):
    f = lambda x: lax.fori_loop(0, 3, lambda i, x: x * 2, x)
    self.assertAllClose(f(2.), 16., check_dtypes=False)
    self.assertAllClose(api.jvp(f, (2.,), (1.,)), (16., 8.), check_dtypes=False)
    jtu.check_grads(f, (2.,), order=2, modes=["fwd"])

    f = lambda x: lax.fori_loop(0, 3, lambda i, x: x * (i + 1), x)
    self.assertAllClose(f(2.), 12., check_dtypes=False)
    self.assertAllClose(api.jvp(f, (2.,), (1.,)), (12., 6.), check_dtypes=False)
    jtu.check_grads(f, (2.,), order=2, modes=["fwd"])

  def testWhileJVPWithGrowingNonzeroTangents(self):
    rng = np.random.RandomState(0)

    def cond(state):
      i, x, y, z = state
      return i < 2

    def body(state):
      i, x, y, z = state
      y = x * x
      z = y * y
      return i + 1, x, y, z

    y, z = rng.randn(2), rng.randn(2)
    def loop(loop_impl, x):
      return loop_impl(cond, body, (0, x, y, z))[1]

    loop_lax = partial(loop, lax.while_loop)
    loop_ref = partial(loop, while_loop_reference)

    x = rng.randn(2)
    ans = api.jvp(loop_lax, (x,), (x,))
    expected = api.jvp(loop_ref, (x,), (x,))
    self.assertAllClose(ans, expected, check_dtypes=False)

    jtu.check_grads(loop_lax, (x,), order=2, modes=["fwd"])

  @parameterized.named_parameters(
      dict(testcase_name="_loop={}".format(loop), loop=loop)
      for loop in ["while", "fori", "fori_inside_cond", "fori_inside_scan"])
  def testWhileGradError(self, loop: str = "fori_inside_scan"):
    # Raise error for vjp for loops
    if loop == "while":
      func = lambda x: lax.while_loop(lambda i: i < 5., lambda i: i + 1., x)
    elif loop == "fori":
      func = lambda x: lax.fori_loop(x, x + 2., lambda i, c: c, x)
    elif loop == "fori_inside_jit":
      func = api.jit(lambda x: lax.fori_loop(x, x + 2., lambda i, c: c, x))
    elif loop == "fori_inside_cond":
      func = lambda x: lax.cond(True, x,
                                lambda x: lax.fori_loop(x, x + 2., lambda i, c: c, x),
                                 1., lambda x: x)
    elif loop == "fori_inside_scan":
      func = lambda x: lax.scan(lambda c, x: (lax.fori_loop(x, x + 2., lambda i, c1: c1 * c, x),
                                              None),
                                x, np.ones(2))[0]
    else:
      assert False

    with self.assertRaisesRegex(ValueError, "Reverse-mode differentiation does not work for lax.while_loop"):
      api.grad(func)(1.)

    api.linearize(func, 1.)  # Linearization works

  def testIssue1316(self):
    def f(carry, _):
      c, key = carry
      key, _ = random.split(key)
      return (c, key), ()

    key = random.PRNGKey(0)
    api.grad(lambda c: lax.scan(f, (c, key), np.ones(3))[0][0])(0.)  # doesn't crash

  def testIssue1361(self):
    @api.jit
    def jit_run_scan(x):
      def fun(carry, _):
        x, _ = carry
        return (2 * x, 0.), None
      (x, _), _ = lax.scan(fun, (x, 0.), jnp.arange(3))
      return x

    api.grad(lambda x: jit_run_scan(x))(0.)  # doesn't crash

  def test_custom_root_scalar(self):

    def scalar_solve(f, y):
      return y / f(1.0)

    def binary_search(func, x0, low=0.0, high=100.0):
      del x0  # unused

      def cond(state):
        low, high = state
        midpoint = 0.5 * (low + high)
        return (low < midpoint) & (midpoint < high)

      def body(state):
        low, high = state
        midpoint = 0.5 * (low + high)
        update_upper = func(midpoint) > 0
        low = jnp.where(update_upper, low, midpoint)
        high = jnp.where(update_upper, midpoint, high)
        return (low, high)

      solution, _ = lax.while_loop(cond, body, (low, high))
      return solution

    def sqrt_cubed(x, tangent_solve=scalar_solve):
      f = lambda y: y ** 2 - x ** 3
      return lax.custom_root(f, 0.0, binary_search, tangent_solve)

    value, grad = api.value_and_grad(sqrt_cubed)(5.0)
    self.assertAllClose(value, 5 ** 1.5, check_dtypes=False, rtol=1e-6)
    self.assertAllClose(grad, api.grad(pow)(5.0, 1.5), check_dtypes=False,
                        rtol=1e-7)
    jtu.check_grads(sqrt_cubed, (5.0,), order=2,
                    rtol={jnp.float32: 1e-2, jnp.float64: 1e-3})

    inputs = jnp.array([4.0, 5.0])
    results = api.vmap(sqrt_cubed)(inputs)
    self.assertAllClose(results, inputs ** 1.5, check_dtypes=False)

    results = api.jit(sqrt_cubed)(5.0)
    self.assertAllClose(results, 5.0 ** 1.5, check_dtypes=False,
                        rtol={np.float64:1e-7})

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def test_custom_root_vector_with_solve_closure(self):

    def vector_solve(f, y):
      return jnp.linalg.solve(api.jacobian(f)(y), y)

    def linear_solve(a, b):
      f = lambda y: high_precision_dot(a, y) - b
      x0 = jnp.zeros_like(b)
      solution = jnp.linalg.solve(a, b)
      oracle = lambda func, x0: solution
      return lax.custom_root(f, x0, oracle, vector_solve)

    rng = np.random.RandomState(0)
    a = rng.randn(2, 2)
    b = rng.randn(2)
    jtu.check_grads(linear_solve, (a, b), order=2,
                    atol={np.float32: 1e-2, np.float64: 1e-11})

    actual = api.jit(linear_solve)(a, b)
    expected = jnp.linalg.solve(a, b)
    self.assertAllClose(expected, actual)

  def test_custom_root_with_custom_linear_solve(self):

    # TODO(shoyer): Figure out why this fails and re-enable it.
    if jtu.device_under_test() == "tpu":
      raise SkipTest("Test fails on TPU")

    def linear_solve(a, b):
      f = lambda x: jnp.dot(a, x) - b
      factors = jsp.linalg.cho_factor(a)
      cho_solve = lambda f, b: jsp.linalg.cho_solve(factors, b)
      def pos_def_solve(g, b):
        return lax.custom_linear_solve(g, b, cho_solve, symmetric=True)
      return lax.custom_root(f, b, cho_solve, pos_def_solve)

    rng = np.random.RandomState(0)
    a = rng.randn(2, 2)
    b = rng.randn(2)

    actual = linear_solve(jnp.dot(a, a.T), b)
    expected = jnp.linalg.solve(jnp.dot(a, a.T), b)
    self.assertAllClose(expected, actual)

    actual = api.jit(linear_solve)(jnp.dot(a, a.T), b)
    expected = jnp.linalg.solve(jnp.dot(a, a.T), b)
    self.assertAllClose(expected, actual)

    jtu.check_grads(lambda x, y: linear_solve(jnp.dot(x, x.T), y),
                    (a, b), order=2, rtol={jnp.float32: 1e-2})

  def test_custom_root_errors(self):
    with self.assertRaisesRegex(TypeError, re.escape("f() output pytree")):
      lax.custom_root(lambda x: (x, x), 0.0, lambda f, x: x, lambda f, x: x)
    with self.assertRaisesRegex(TypeError, re.escape("solve() output pytree")):
      lax.custom_root(lambda x: x, 0.0, lambda f, x: (x, x), lambda f, x: x)

    def dummy_root_usage(x):
      f = lambda y: x - y
      return lax.custom_root(f, 0.0, lambda f, x: x, lambda f, x: (x, x))

    with self.assertRaisesRegex(
        TypeError, re.escape("tangent_solve() output pytree")):
      api.jvp(dummy_root_usage, (0.0,), (0.0,))

  @parameterized.named_parameters(
      {"testcase_name": "nonsymmetric", "symmetric": False},
      {"testcase_name": "symmetric", "symmetric": True},
  )
  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def test_custom_linear_solve(self, symmetric):

    def explicit_jacobian_solve(matvec, b):
      return lax.stop_gradient(jnp.linalg.solve(api.jacobian(matvec)(b), b))

    def matrix_free_solve(matvec, b):
      return lax.custom_linear_solve(
          matvec, b, explicit_jacobian_solve, explicit_jacobian_solve,
          symmetric=symmetric)

    def linear_solve(a, b):
      return matrix_free_solve(partial(high_precision_dot, a), b)

    rng = np.random.RandomState(0)
    a = rng.randn(3, 3)
    if symmetric:
      a = a + a.T
    b = rng.randn(3)
    jtu.check_grads(linear_solve, (a, b), order=2, rtol=2e-3)

    expected = jnp.linalg.solve(a, b)
    actual = api.jit(linear_solve)(a, b)
    self.assertAllClose(expected, actual)

    c = rng.randn(3, 2)
    expected = jnp.linalg.solve(a, c)
    actual = api.vmap(linear_solve, (None, 1), 1)(a, c)
    self.assertAllClose(expected, actual)

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def test_custom_linear_solve_zeros(self):
    def explicit_jacobian_solve(matvec, b):
      return lax.stop_gradient(jnp.linalg.solve(api.jacobian(matvec)(b), b))

    def matrix_free_solve(matvec, b):
      return lax.custom_linear_solve(matvec, b, explicit_jacobian_solve,
                                     explicit_jacobian_solve)

    def linear_solve(a, b):
      return matrix_free_solve(partial(high_precision_dot, a), b)

    rng = np.random.RandomState(0)
    a = rng.randn(3, 3)
    b = rng.randn(3)
    jtu.check_grads(lambda x: linear_solve(x, b), (a,), order=2,
                    rtol={np.float32: 5e-3})
    jtu.check_grads(lambda x: linear_solve(a, x), (b,), order=2,
                    rtol={np.float32: 5e-3})

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def test_custom_linear_solve_iterative(self):

    def richardson_iteration(matvec, b, omega=0.1, tolerance=1e-6):
      # Equivalent to vanilla gradient descent:
      # https://en.wikipedia.org/wiki/Modified_Richardson_iteration
      def cond(x):
        return jnp.linalg.norm(matvec(x) - b) > tolerance
      def body(x):
        return x + omega * (b - matvec(x))
      return lax.while_loop(cond, body, b)

    def matrix_free_solve(matvec, b):
      return lax.custom_linear_solve(matvec, b, richardson_iteration,
                                     richardson_iteration)

    def build_and_solve(a, b):
      # intentionally non-linear in a and b
      matvec = partial(high_precision_dot, jnp.exp(a))
      return matrix_free_solve(matvec, jnp.cos(b))

    rng = np.random.RandomState(0)
    a = rng.randn(2, 2)
    b = rng.randn(2)
    expected = jnp.linalg.solve(jnp.exp(a), jnp.cos(b))
    actual = build_and_solve(a, b)
    self.assertAllClose(expected, actual, atol=1e-5)
    jtu.check_grads(build_and_solve, (a, b), atol=1e-5, order=2,
                    rtol={jnp.float32: 6e-2, jnp.float64: 2e-3})

    # vmap across an empty dimension
    jtu.check_grads(
        api.vmap(build_and_solve), (a[None, :, :], b[None, :]),
        atol=1e-5,
        order=2,
        rtol={jnp.float32: 6e-2, jnp.float64: 2e-3})

  def test_custom_linear_solve_cholesky(self):

    def positive_definite_solve(a, b):
      factors = jsp.linalg.cho_factor(a)
      def solve(matvec, x):
        return jsp.linalg.cho_solve(factors, x)
      matvec = partial(high_precision_dot, a)
      return lax.custom_linear_solve(matvec, b, solve, symmetric=True)

    rng = np.random.RandomState(0)
    a = rng.randn(2, 2)
    b = rng.randn(2)

    expected = jnp.linalg.solve(np.asarray(posify(a)), b)
    actual = positive_definite_solve(posify(a), b)
    self.assertAllClose(expected, actual)

    actual = api.jit(positive_definite_solve)(posify(a), b)
    self.assertAllClose(expected, actual)

    # numerical gradients are only well defined if ``a`` is guaranteed to be
    # positive definite.
    jtu.check_grads(
        lambda x, y: positive_definite_solve(posify(x), y),
        (a, b), order=2, rtol=1e-2)

  def test_custom_linear_solve_complex(self):

    def solve(a, b):
      def solve(matvec, x):
        return jsp.linalg.solve(a, x)
      def tr_solve(matvec, x):
        return jsp.linalg.solve(a.T, x)
      matvec = partial(high_precision_dot, a)
      return lax.custom_linear_solve(matvec, b, solve, tr_solve)

    rng = np.random.RandomState(0)
    a = 0.5 * rng.randn(2, 2) + 0.5j * rng.randn(2, 2)
    b = 0.5 * rng.randn(2) + 0.5j * rng.randn(2)
    jtu.check_grads(solve, (a, b), order=2, rtol=1e-2)

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def test_custom_linear_solve_lu(self):

    # TODO(b/143528110): re-enable when underlying XLA TPU issue is fixed
    if jtu.device_under_test() == "tpu":
      raise SkipTest("Test fails on TPU")

    def linear_solve(a, b):
      a_factors = jsp.linalg.lu_factor(a)
      at_factors = jsp.linalg.lu_factor(a.T)
      def solve(matvec, x):
        return jsp.linalg.lu_solve(a_factors, x)
      def transpose_solve(vecmat, x):
        return jsp.linalg.lu_solve(at_factors, x)
      return lax.custom_linear_solve(
          partial(high_precision_dot, a), b, solve, transpose_solve)

    rng = np.random.RandomState(0)
    a = rng.randn(3, 3)
    b = rng.randn(3)

    expected = jnp.linalg.solve(a, b)
    actual = linear_solve(a, b)
    self.assertAllClose(expected, actual)

    jtu.check_grads(linear_solve, (a, b), order=2, rtol=2e-3)

    # regression test for https://github.com/google/jax/issues/1536
    jtu.check_grads(api.jit(linear_solve), (a, b), order=2,
                    rtol={np.float32: 2e-3})

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def test_custom_linear_solve_without_transpose_solve(self):

    def explicit_jacobian_solve(matvec, b):
      return lax.stop_gradient(jnp.linalg.solve(api.jacobian(matvec)(b), b))

    def loss(a, b):
      matvec = partial(high_precision_dot, a)
      x = lax.custom_linear_solve(matvec, b, explicit_jacobian_solve)
      return jnp.sum(x)

    rng = np.random.RandomState(0)
    a = rng.randn(2, 2)
    b = rng.randn(2)

    jtu.check_grads(loss, (a, b), order=2, modes=['fwd'],
                    atol={np.float32: 2e-3, np.float64: 1e-11})
    jtu.check_grads(api.vmap(loss), (a[None,:,:], b[None,:]), order=2,
                    modes=['fwd'], atol={np.float32: 2e-3, np.float64: 1e-11})

    with self.assertRaisesRegex(TypeError, "transpose_solve required"):
      api.grad(loss)(a, b)

  @jtu.skip_on_flag("jax_skip_slow_tests", True)
  def test_custom_linear_solve_pytree(self):
    """Test custom linear solve with inputs and outputs that are pytrees."""

    def unrolled_matvec(mat, x):
      """Apply a Python list of lists of scalars to a list of scalars."""
      result = []
      for i in range(len(mat)):
        v = 0
        for j in range(len(x)):
          if mat[i][j] is not None:
            v += mat[i][j] * x[j]
        result.append(v)
      return result

    def unrolled_substitution_solve(matvec, b, lower_tri):
      """Solve a triangular unrolled system with fwd/back substitution."""
      zero = jnp.zeros(())
      one = jnp.ones(())
      x = [zero for _ in b]
      ordering = range(len(b)) if lower_tri else range(len(b) - 1, -1, -1)
      for i in ordering:
        residual = b[i] - matvec(x)[i]
        diagonal = matvec([one if i == j else zero for j in range(len(b))])[i]
        x[i] = residual / diagonal
      return x

    def custom_unrolled_lower_tri_solve(mat, b):
      return lax.custom_linear_solve(
          partial(unrolled_matvec, mat), b,
          partial(unrolled_substitution_solve, lower_tri=True),
          partial(unrolled_substitution_solve, lower_tri=False))

    mat = [[1.0, None, None, None, None, None, None],
           [1.0, 1.0, None, None, None, None, None],
           [None, 1.0, 1.0, None, None, None, None],
           [None, None, 1.0, 1.0, None, None, None],
           [None, None, None, 1.0, 1.0, None, None],
           [None, None, None, None, None, 2.0, None],
           [None, None, None, None, None, 4.0, 3.0]]

    rng = np.random.RandomState(0)
    b = list(rng.randn(7))

    # Non-batched
    jtu.check_grads(custom_unrolled_lower_tri_solve, (mat, b), order=2,
                    rtol={jnp.float32: 2e-2})

    # Batch one element of b (which, because of unrolling, should only affect
    # the first block of outputs)
    b_bat = list(b)
    b_bat[3] = rng.randn(3)
    jtu.check_grads(
        api.vmap(
            custom_unrolled_lower_tri_solve,
            in_axes=(None, [None, None, None, 0, None, None, None]),
            out_axes=[0, 0, 0, 0, 0, None, None]), (mat, b_bat),
        order=2,
        rtol={jnp.float32: 1e-2})

    # Batch one element of mat (again only affecting first block)
    mat[2][1] = rng.randn(3)
    mat_axis_tree = [
        [0 if i == 2 and j == 1 else None for j in range(7)] for i in range(7)
    ]
    jtu.check_grads(
        api.vmap(
            custom_unrolled_lower_tri_solve,
            in_axes=(mat_axis_tree, None),
            out_axes=[0, 0, 0, 0, 0, None, None]), (mat, b),
        order=2)

  def test_custom_linear_solve_errors(self):

    solve = lambda f, x: x

    with self.assertRaisesRegex(TypeError, re.escape("matvec() output pytree")):
      lax.custom_linear_solve(lambda x: [x], 1.0, solve, solve)
    with self.assertRaisesRegex(TypeError, re.escape("solve() output pytree")):
      lax.custom_linear_solve(lambda x: x, 1.0, lambda f, x: [x], solve)
    with self.assertRaisesRegex(
        TypeError, re.escape("transpose_solve() output pytree")):
      lax.custom_linear_solve(lambda x: x, 1.0, solve, lambda f, x: [x])

    with self.assertRaisesRegex(ValueError, re.escape("solve() output shapes")):
      lax.custom_linear_solve(lambda x: x, 1.0, lambda f, x: jnp.ones(2), solve)

    def bad_matvec_usage(a):
      return lax.custom_linear_solve(
          lambda x: a * jnp.ones(2), 1.0, solve, solve)
    with self.assertRaisesRegex(ValueError, re.escape("matvec() output shapes")):
      api.jvp(bad_matvec_usage, (1.0,), (1.0,))

  def testIssue810(self):
    def loss(A):
      def step(x, i):
        return jnp.matmul(A, x), None
      init_x = jnp.zeros(A.shape[-1:])
      last_x, _ = lax.scan(step, init_x, jnp.arange(10))
      return jnp.sum(last_x)

    A = jnp.zeros((3, 3))
    # The second DUS was unnecessarily replicating A across time.
    # We check XLA because _scan_impl is "underneath" the jaxpr language.
    s = str(api.xla_computation(api.grad(loss))(A).as_hlo_text())
    assert s.count("dynamic-update-slice(") < 2

  def testScanLengthArg(self):
    def arange(n):
      return lax.scan(lambda c, _: (c + 1, c), 0, None, length=n)[1]

    ans = arange(10)
    expected = np.arange(10)
    self.assertAllClose(ans, expected, check_dtypes=False)

  def test_while_loop_of_pmap(self):
    # code from jsnoek@

    def body(i, x):
      result = api.pmap(lambda z: lax.psum(jnp.sin(z), 'i'), axis_name='i')(x)
      return result + x
    f_loop = lambda x: lax.fori_loop(0, 3, body, x)  # noqa: F821
    ans = f_loop(jnp.ones(api.device_count()))
    del body, f_loop

    def body2(i, x):
      result = jnp.broadcast_to(jnp.sin(x).sum(), x.shape)
      return result + x
    g_loop = lambda x: lax.fori_loop(0, 3, body2, x)
    expected = g_loop(jnp.ones(api.device_count()))

    self.assertAllClose(ans, expected, check_dtypes=False)

  def test_while_loop_of_pmap_error_message(self):

    def body(i, x):
      result = api.pmap(lambda z: lax.psum(jnp.sin(z), 'i'), axis_name='i')(x)
      return result + x
    f_loop = lambda x: lax.fori_loop(0, 3, body, x)

    too_big = 2 * api.device_count()

    self.assertRaisesRegex(
        ValueError,
        re.escape(
            "compiling a primitive computation `while` that requires {} "
            "replicas, but only {} XLA devices are available on backend {}."
            .format(too_big, api.device_count(), jtu.device_under_test())),
        lambda: f_loop(jnp.ones(too_big)))

  @parameterized.named_parameters(
      {"testcase_name": "_{}".format(scan_name),
       "scan": scan_impl}
      for scan_impl, scan_name in SCAN_IMPLS)
  def test_scan_reverse(self, scan):
    def cumsum(x, reverse):
      return scan(lambda c, x: (c + x, c + x), 0, x, reverse=reverse)[1]

    x = np.array([3, 1, 4, 1, 5, 9])
    self.assertAllClose(np.cumsum(x), cumsum(x, False), check_dtypes=False)
    self.assertAllClose(np.cumsum(x[::-1])[::-1], cumsum(x, True), check_dtypes=False)

    with api.disable_jit():
      self.assertAllClose(np.cumsum(x), cumsum(x, False), check_dtypes=False)
    with api.disable_jit():
      self.assertAllClose(np.cumsum(x[::-1])[::-1], cumsum(x, True), check_dtypes=False)

  def test_scan_unroll(self):
    d = jnp.ones(2)
    def f(c, a):
      assert a.shape == (3,)
      assert c.shape == (4,)
      b = jnp.cos(jnp.sum(jnp.sin(a)) + jnp.sum(jnp.cos(c)) + jnp.sum(jnp.tan(d)))
      c = jnp.sin(c * b)
      assert b.shape == ()
      return c, b

    xs = jnp.ones((5, 3))
    c = jnp.ones(4)

    scan = lambda c, xs: lax.scan(f, c, xs)
    scan_unrolled = lambda c, xs: lax.scan(f, c, xs, unroll=2)

    # jaxprs should be the same size
    self.assertEqual(
        len(str(api.make_jaxpr(scan)(c, xs))),
        len(str(api.make_jaxpr(scan_unrolled)(c, xs))))

    # but HLO should grow due to unrolling
    self.assertLess(
        len(str(api.xla_computation(scan)(c, xs).as_hlo_text())),
        len(str(api.xla_computation(scan_unrolled)(c, xs).as_hlo_text())))

  def test_disable_jit_cond_with_vmap(self):
    # https://github.com/google/jax/issues/3093
    def fn(t):
      return lax.cond(t > 0, 0, lambda x: 0, 0, lambda x: 1)
    fn = api.vmap(fn)

    with api.disable_jit():
      _ = fn(jnp.array([1]))  # doesn't crash

  def test_disable_jit_while_loop_with_vmap(self):
    # https://github.com/google/jax/issues/2823
    def trivial_while(y):
      return lax.while_loop(lambda x: x < 10.0, lambda x: x + 1.0, y)
    with api.disable_jit():
      api.vmap(trivial_while)(jnp.array([3.0,4.0]))  # doesn't crash

  def test_vmaps_of_while_loop(self):
    # https://github.com/google/jax/issues/3164
    def f(x, n): return lax.fori_loop(0, n, lambda _, x: x + 1, x)
    x, n = jnp.arange(3), jnp.arange(4)
    api.vmap(api.vmap(f, (None, 0)), (0, None))(x, n)  # doesn't crash


  @parameterized.named_parameters(
      {"testcase_name": f"_{shape}_axis={axis}",
       "shape": shape, "axis": axis}
      for shape in [
        [0], [1], [2], [3], [5], [10], [1000],
        [2, 3], [7, 5], [5, 6, 7]
      ]
      for axis in range(-len(shape), len(shape) - 1))
  def testAssociativeScanUnstructured(self, shape, axis):
    data = np.arange(np.prod(shape)).reshape(shape) + 7
    expected = np.cumsum(data, axis=axis)
    result = lax.associative_scan(operator.add, data, axis=axis)
    self.assertAllClose(result, expected, check_dtypes=False)

  def testAssociativeScanUnstructured1000Reverse(self):
    data = np.arange(1000) + 32
    expected = np.cumsum(data[::-1])[::-1]
    result = lax.associative_scan(operator.add, data, reverse=True)
    self.assertAllClose(result, expected, check_dtypes=False)

  def testAssociativeScanStructured3(self):
    pair = collections.namedtuple('pair', ('first', 'second'))
    data = pair(first=np.array([0., 1., 2.]),
                second=np.array([0., 10., 20.]))

    def fn(a, b):
      return pair(first=a.first + b.first,
                  second=a.second + b.second)

    result = lax.associative_scan(fn, elems=data)
    self.assertAllClose(result.first, np.array([0., 1., 3.]),
                        check_dtypes=False)
    self.assertAllClose(result.second, np.array([0., 10., 30.]),
                        check_dtypes=False)

  def test_scan_typecheck_param(self):
    d = jnp.ones(2)
    def f(c, a):
      b = jnp.cos(jnp.sum(a) + jnp.sum(c) + jnp.sum(d))
      c = jnp.sin(c * b)
      return c, b

    xs = jnp.ones((5, 3))
    c = jnp.ones(4)
    scan_fun = lambda c, xs: lax.scan(f, c, xs)

    def new_jaxpr():
      jaxpr = api.make_jaxpr(scan_fun)(c, xs).jaxpr
      scan = next(eqn for eqn in jaxpr.eqns if eqn.primitive.name == 'scan')
      return jaxpr, scan

    jaxpr, eqn = new_jaxpr()
    eqn.params['reverse'] = 4
    self.assertRaisesRegex(
        core.JaxprTypeError,
        re.escape('invalid scan param reverse of type int, bool required: 4'),
        lambda: core.check_jaxpr(jaxpr))

    jaxpr, eqn = new_jaxpr()
    eqn.params['num_consts'] = -3
    self.assertRaisesRegex(
        core.JaxprTypeError,
        re.escape('invalid scan param num_consts of type int, '
                  'non-negative int required: -3'),
        lambda: core.check_jaxpr(jaxpr))

  def test_cond_typecheck_param(self):
    def new_jaxpr():
      jaxpr = api.make_jaxpr(
          lambda x: lax.switch(0, [jnp.sin, jnp.cos], x))(1.).jaxpr
      cond = next(eqn for eqn in jaxpr.eqns if eqn.primitive.name == 'cond')
      return jaxpr, cond

    jaxpr, eqn = new_jaxpr()
    eqn.params['branches'] = (4, 2)
    self.assertRaisesRegex(
        core.JaxprTypeError,
        re.escape('invalid cond param branches of type tuple, '
                  'tuple of ClosedJaxpr required: (4, 2)'),
        lambda: core.check_jaxpr(jaxpr))

    jaxpr, eqn = new_jaxpr()
    eqn.params['linear'] = (4, 2)
    self.assertRaisesRegex(
        core.JaxprTypeError,
        re.escape('invalid cond param linear of type tuple, '
                  'tuple of bool required: (4, 2)'),
        lambda: core.check_jaxpr(jaxpr))

    jaxpr, eqn = new_jaxpr()
    eqn.params['linear'] = 'multi\nline'
    self.assertRaisesRegex(
        core.JaxprTypeError,
        r'invalid cond param linear of type str, '
        r'tuple of bool required:\nmulti\nline',
        lambda: core.check_jaxpr(jaxpr))

  @parameterized.named_parameters(
      {"testcase_name": f"_dtype={dtype.__name__}", "dtype": dtype}
      for dtype in jtu.dtypes.all_integer)
  def test_scan_init_weak_type(self, dtype):
    def func(carry, x):
      return carry + x, x
    init_weak = 0  # Python scalars are weakly-typed.
    x = jnp.ones(5, dtype=dtype)
    carry, result = lax.scan(func, init_weak, x)
    self.assertEqual(carry, x.sum())
    self.assertArraysEqual(result, x)

  @parameterized.named_parameters(
      {"testcase_name": f"_dtype={dtype.__name__}", "dtype": dtype}
      for dtype in jtu.dtypes.all_integer)
  def test_while_loop_init_weak_type(self, dtype):
    # This tests whether lax.while_loop can properly handle weakly-typed
    # initial values.
    def cond_fun(val):
      return val < 2
    def body_fun(val):
      return val + increment
    increment = jnp.array(1, dtype=dtype)
    init_weak = 0  # Python scalars are weakly-typed.
    result = lax.while_loop(cond_fun, body_fun, init_weak)
    self.assertArraysEqual(result, jnp.full_like(increment, 2))

  def test_scan_vjp_forwards_extensive_residuals(self):
    # https://github.com/google/jax/issues/4510
    def cumprod(x):
      s = jnp.ones((2, 32), jnp.float32)
      return lax.scan(lambda s, x: (x*s, s), s, x)

    rng = np.random.RandomState(1234)
    x = jnp.asarray(rng.randn(32, 2, 32).astype('float32'))
    _, vjp_fun = api.vjp(cumprod, x)

    # Need to spelunk into vjp_fun. This is fragile, and if it causes problems
    # just skip this test.
    *_, ext_res = vjp_fun.args[0].args[0]
    self.assertIs(ext_res, x)

    x = rng.randn(32, 2, 32).astype('float32')  # numpy.ndarray, not DeviceArray
    _, vjp_fun = api.vjp(cumprod, x)
    *_, ext_res = vjp_fun.args[0].args[0]
    self.assertIsInstance(ext_res, xla.DeviceArray)

  @skipIf(not config.omnistaging_enabled,
          "vmap collectives only supported when omnistaging is enabled")
  def test_scan_vmap_collectives(self):
    def scan_f(state, x):
      s = lax.psum(state, 'i') * x
      return state, s

    def scan(state, xs):
      return lax.scan(scan_f, state, xs)

    scan_v = api.vmap(scan, in_axes=0, out_axes=0, axis_name='i')
    self.assertAllClose(
      scan_v(jnp.ones([1]), jnp.arange(5).reshape((1, 5))),
      (jnp.array([1.]), jnp.array([[0., 1., 2., 3., 4.]])))


if __name__ == '__main__':
  absltest.main(testLoader=jtu.JaxTestLoader())