#
# Copyright (C) 2014 Intel Corporation
#
# Permission is hereby granted, free of charge, to any person obtaining a
# copy of this software and associated documentation files (the "Software"),
# to deal in the Software without restriction, including without limitation
# the rights to use, copy, modify, merge, publish, distribute, sublicense,
# and/or sell copies of the Software, and to permit persons to whom the
# Software is furnished to do so, subject to the following conditions:
#
# The above copyright notice and this permission notice (including the next
# paragraph) shall be included in all copies or substantial portions of the
# Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
# FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
# IN THE SOFTWARE.
#
# Authors:
#    Jason Ekstrand (jason@jlekstrand.net)

from __future__ import print_function

from collections import OrderedDict
import nir_algebraic
from nir_opcodes import type_sizes
import itertools
import struct
from math import pi

# Convenience variables
a = 'a'
b = 'b'
c = 'c'
d = 'd'
e = 'e'

# Written in the form (<search>, <replace>) where <search> is an expression
# and <replace> is either an expression or a value.  An expression is
# defined as a tuple of the form ([~]<op>, <src0>, <src1>, <src2>, <src3>)
# where each source is either an expression or a value.  A value can be
# either a numeric constant or a string representing a variable name.
#
# If the opcode in a search expression is prefixed by a '~' character, this
# indicates that the operation is inexact.  Such operations will only get
# applied to SSA values that do not have the exact bit set.  This should be
# used by by any optimizations that are not bit-for-bit exact.  It should not,
# however, be used for backend-requested lowering operations as those need to
# happen regardless of precision.
#
# Variable names are specified as "[#]name[@type][(cond)][.swiz]" where:
# "#" indicates that the given variable will only match constants,
# type indicates that the given variable will only match values from ALU
#    instructions with the given output type,
# (cond) specifies an additional condition function (see nir_search_helpers.h),
# swiz is a swizzle applied to the variable (only in the <replace> expression)
#
# For constants, you have to be careful to make sure that it is the right
# type because python is unaware of the source and destination types of the
# opcodes.
#
# All expression types can have a bit-size specified.  For opcodes, this
# looks like "op@32", for variables it is "a@32" or "a@uint32" to specify a
# type and size.  In the search half of the expression this indicates that it
# should only match that particular bit-size.  In the replace half of the
# expression this indicates that the constructed value should have that
# bit-size.
#
# If the opcode in a replacement expression is prefixed by a '!' character,
# this indicated that the new expression will be marked exact.
#
# A special condition "many-comm-expr" can be used with expressions to note
# that the expression and its subexpressions have more commutative expressions
# than nir_replace_instr can handle.  If this special condition is needed with
# another condition, the two can be separated by a comma (e.g.,
# "(many-comm-expr,is_used_once)").

# based on https://web.archive.org/web/20180105155939/http://forum.devmaster.net/t/fast-and-accurate-sine-cosine/9648
def lowered_sincos(c):
    x = ('fsub', ('fmul', 2.0, ('ffract', ('fadd', ('fmul', 0.5 / pi, a), c))), 1.0)
    x = ('fmul', ('fsub', x, ('fmul', x, ('fabs', x))), 4.0)
    return ('ffma', ('ffma', x, ('fabs', x), ('fneg', x)), 0.225, x)

def intBitsToFloat(i):
    return struct.unpack('!f', struct.pack('!I', i))[0]

optimizations = [

   (('imul', a, '#b@32(is_pos_power_of_two)'), ('ishl', a, ('find_lsb', b)), '!options->lower_bitops'),
   (('imul', a, '#b@32(is_neg_power_of_two)'), ('ineg', ('ishl', a, ('find_lsb', ('iabs', b)))), '!options->lower_bitops'),
   (('ishl', a, '#b@32'), ('imul', a, ('ishl', 1, b)), 'options->lower_bitops'),

   (('unpack_64_2x32_split_x', ('imul_2x32_64(is_used_once)', a, b)), ('imul', a, b)),
   (('unpack_64_2x32_split_x', ('umul_2x32_64(is_used_once)', a, b)), ('imul', a, b)),
   (('imul_2x32_64', a, b), ('pack_64_2x32_split', ('imul', a, b), ('imul_high', a, b)), 'options->lower_mul_2x32_64'),
   (('umul_2x32_64', a, b), ('pack_64_2x32_split', ('imul', a, b), ('umul_high', a, b)), 'options->lower_mul_2x32_64'),
   (('udiv', a, 1), a),
   (('idiv', a, 1), a),
   (('umod', a, 1), 0),
   (('imod', a, 1), 0),
   (('udiv', a, '#b@32(is_pos_power_of_two)'), ('ushr', a, ('find_lsb', b)), '!options->lower_bitops'),
   (('idiv', a, '#b@32(is_pos_power_of_two)'), ('imul', ('isign', a), ('ushr', ('iabs', a), ('find_lsb', b))), 'options->lower_idiv'),
   (('idiv', a, '#b@32(is_neg_power_of_two)'), ('ineg', ('imul', ('isign', a), ('ushr', ('iabs', a), ('find_lsb', ('iabs', b))))), 'options->lower_idiv'),
   (('umod', a, '#b(is_pos_power_of_two)'),    ('iand', a, ('isub', b, 1))),

   (('~fneg', ('fneg', a)), a),
   (('ineg', ('ineg', a)), a),
   (('fabs', ('fneg', a)), ('fabs', a)),
   (('fabs', ('u2f', a)), ('u2f', a)),
   (('iabs', ('iabs', a)), ('iabs', a)),
   (('iabs', ('ineg', a)), ('iabs', a)),
   (('f2b', ('fneg', a)), ('f2b', a)),
   (('i2b', ('ineg', a)), ('i2b', a)),
   (('~fadd', a, 0.0), a),
   (('iadd', a, 0), a),
   (('usadd_4x8', a, 0), a),
   (('usadd_4x8', a, ~0), ~0),
   (('~fadd', ('fmul', a, b), ('fmul', a, c)), ('fmul', a, ('fadd', b, c))),
   (('iadd', ('imul', a, b), ('imul', a, c)), ('imul', a, ('iadd', b, c))),
   (('~fadd', ('fneg', a), a), 0.0),
   (('iadd', ('ineg', a), a), 0),
   (('iadd', ('ineg', a), ('iadd', a, b)), b),
   (('iadd', a, ('iadd', ('ineg', a), b)), b),
   (('~fadd', ('fneg', a), ('fadd', a, b)), b),
   (('~fadd', a, ('fadd', ('fneg', a), b)), b),
   (('fadd', ('fsat', a), ('fsat', ('fneg', a))), ('fsat', ('fabs', a))),
   (('~fmul', a, 0.0), 0.0),
   (('imul', a, 0), 0),
   (('umul_unorm_4x8', a, 0), 0),
   (('umul_unorm_4x8', a, ~0), a),
   (('~fmul', a, 1.0), a),
   (('imul', a, 1), a),
   (('fmul', a, -1.0), ('fneg', a)),
   (('imul', a, -1), ('ineg', a)),
   # If a < 0: fsign(a)*a*a => -1*a*a => -a*a => abs(a)*a
   # If a > 0: fsign(a)*a*a => 1*a*a => a*a => abs(a)*a
   # If a == 0: fsign(a)*a*a => 0*0*0 => abs(0)*0
   (('fmul', ('fsign', a), ('fmul', a, a)), ('fmul', ('fabs', a), a)),
   (('fmul', ('fmul', ('fsign', a), a), a), ('fmul', ('fabs', a), a)),
   (('~ffma', 0.0, a, b), b),
   (('~ffma', a, b, 0.0), ('fmul', a, b)),
   (('ffma', 1.0, a, b), ('fadd', a, b)),
   (('ffma', -1.0, a, b), ('fadd', ('fneg', a), b)),
   (('~flrp', a, b, 0.0), a),
   (('~flrp', a, b, 1.0), b),
   (('~flrp', a, a, b), a),
   (('~flrp', 0.0, a, b), ('fmul', a, b)),

   # flrp(a, a + b, c) => a + flrp(0, b, c) => a + (b * c)
   (('~flrp', a, ('fadd(is_used_once)', a, b), c), ('fadd', ('fmul', b, c), a)),
   (('~flrp@32', a, ('fadd', a, b), c), ('fadd', ('fmul', b, c), a), 'options->lower_flrp32'),
   (('~flrp@64', a, ('fadd', a, b), c), ('fadd', ('fmul', b, c), a), 'options->lower_flrp64'),

   (('~flrp@32', ('fadd', a, b), ('fadd', a, c), d), ('fadd', ('flrp', b, c, d), a), 'options->lower_flrp32'),
   (('~flrp@64', ('fadd', a, b), ('fadd', a, c), d), ('fadd', ('flrp', b, c, d), a), 'options->lower_flrp64'),

   (('~flrp@32', a, ('fmul(is_used_once)', a, b), c), ('fmul', ('flrp', 1.0, b, c), a), 'options->lower_flrp32'),
   (('~flrp@64', a, ('fmul(is_used_once)', a, b), c), ('fmul', ('flrp', 1.0, b, c), a), 'options->lower_flrp64'),

   (('~flrp', ('fmul(is_used_once)', a, b), ('fmul(is_used_once)', a, c), d), ('fmul', ('flrp', b, c, d), a)),

   (('~flrp', a, b, ('b2f', 'c@1')), ('bcsel', c, b, a), 'options->lower_flrp32'),
   (('~flrp', a, 0.0, c), ('fadd', ('fmul', ('fneg', a), c), a)),
   (('ftrunc', a), ('bcsel', ('flt', a, 0.0), ('fneg', ('ffloor', ('fabs', a))), ('ffloor', ('fabs', a))), 'options->lower_ftrunc'),
   (('ffloor', a), ('fsub', a, ('ffract', a)), 'options->lower_ffloor'),
   (('fadd', a, ('fneg', ('ffract', a))), ('ffloor', a), '!options->lower_ffloor'),
   (('ffract', a), ('fsub', a, ('ffloor', a)), 'options->lower_ffract'),
   (('fceil', a), ('fneg', ('ffloor', ('fneg', a))), 'options->lower_fceil'),
   (('~fadd',    ('fmul', a,          ('fadd', 1.0, ('fneg', ('b2f', 'c@1')))), ('fmul', b, ('b2f',  c))), ('bcsel', c, b, a), 'options->lower_flrp32'),
   (('~fadd@32', ('fmul', a,          ('fadd', 1.0, ('fneg',          c   ) )), ('fmul', b,          c )), ('flrp', a, b, c), '!options->lower_flrp32'),
   (('~fadd@64', ('fmul', a,          ('fadd', 1.0, ('fneg',          c   ) )), ('fmul', b,          c )), ('flrp', a, b, c), '!options->lower_flrp64'),
   # These are the same as the previous three rules, but it depends on
   # 1-fsat(x) <=> fsat(1-x).  See below.
   (('~fadd@32', ('fmul', a, ('fsat', ('fadd', 1.0, ('fneg',          c   )))), ('fmul', b, ('fsat', c))), ('flrp', a, b, ('fsat', c)), '!options->lower_flrp32'),
   (('~fadd@64', ('fmul', a, ('fsat', ('fadd', 1.0, ('fneg',          c   )))), ('fmul', b, ('fsat', c))), ('flrp', a, b, ('fsat', c)), '!options->lower_flrp64'),

   (('~fadd', a, ('fmul', ('b2f', 'c@1'), ('fadd', b, ('fneg', a)))), ('bcsel', c, b, a), 'options->lower_flrp32'),
   (('~fadd@32', a, ('fmul',         c , ('fadd', b, ('fneg', a)))), ('flrp', a, b, c), '!options->lower_flrp32'),
   (('~fadd@64', a, ('fmul',         c , ('fadd', b, ('fneg', a)))), ('flrp', a, b, c), '!options->lower_flrp64'),
   (('ffma', a, b, c), ('fadd', ('fmul', a, b), c), 'options->lower_ffma'),
   (('~fadd', ('fmul', a, b), c), ('ffma', a, b, c), 'options->fuse_ffma'),

   (('~fmul', ('fadd', ('iand', ('ineg', ('b2i32', 'a@bool')), ('fmul', b, c)), '#d'), '#e'),
    ('bcsel', a, ('fmul', ('fadd', ('fmul', b, c), d), e), ('fmul', d, e))),

   (('fdph', a, b), ('fdot4', ('vec4', 'a.x', 'a.y', 'a.z', 1.0), b), 'options->lower_fdph'),

   (('fdot4', ('vec4', a, b,   c,   1.0), d), ('fdph',  ('vec3', a, b, c), d), '!options->lower_fdph'),
   (('fdot4', ('vec4', a, 0.0, 0.0, 0.0), b), ('fmul', a, b)),
   (('fdot4', ('vec4', a, b,   0.0, 0.0), c), ('fdot2', ('vec2', a, b), c)),
   (('fdot4', ('vec4', a, b,   c,   0.0), d), ('fdot3', ('vec3', a, b, c), d)),

   (('fdot3', ('vec3', a, 0.0, 0.0), b), ('fmul', a, b)),
   (('fdot3', ('vec3', a, b,   0.0), c), ('fdot2', ('vec2', a, b), c)),

   (('fdot2', ('vec2', a, 0.0), b), ('fmul', a, b)),
   (('fdot2', a, 1.0), ('fadd', 'a.x', 'a.y')),

   # Lower fdot to fsum when it is available
   (('fdot2', a, b), ('fsum2', ('fmul', a, b)), 'options->lower_fdot'),
   (('fdot3', a, b), ('fsum3', ('fmul', a, b)), 'options->lower_fdot'),
   (('fdot4', a, b), ('fsum4', ('fmul', a, b)), 'options->lower_fdot'),
   (('fsum2', a), ('fadd', 'a.x', 'a.y'), 'options->lower_fdot'),

   # If x >= 0 and x <= 1: fsat(1 - x) == 1 - fsat(x) trivially
   # If x < 0: 1 - fsat(x) => 1 - 0 => 1 and fsat(1 - x) => fsat(> 1) => 1
   # If x > 1: 1 - fsat(x) => 1 - 1 => 0 and fsat(1 - x) => fsat(< 0) => 0
   (('~fadd', ('fneg(is_used_once)', ('fsat(is_used_once)', 'a(is_not_fmul)')), 1.0), ('fsat', ('fadd', 1.0, ('fneg', a)))),

   # 1 - ((1 - a) * (1 - b))
   # 1 - (1 - a - b + a*b)
   # 1 - 1 + a + b - a*b
   # a + b - a*b
   # a + b*(1 - a)
   # b*(1 - a) + 1*a
   # flrp(b, 1, a)
   (('~fadd@32', 1.0, ('fneg', ('fmul', ('fadd', 1.0, ('fneg', a)), ('fadd', 1.0, ('fneg', b))))),
    ('flrp', b, 1.0, a), '!options->lower_flrp32'),

   # (a * #b + #c) << #d
   # ((a * #b) << #d) + (#c << #d)
   # (a * (#b << #d)) + (#c << #d)
   (('ishl', ('iadd', ('imul', a, '#b'), '#c'), '#d'),
    ('iadd', ('imul', a, ('ishl', b, d)), ('ishl', c, d))),

   # (a * #b) << #c
   # a * (#b << #c)
   (('ishl', ('imul', a, '#b'), '#c'), ('imul', a, ('ishl', b, c))),
]

# Care must be taken here.  Shifts in NIR uses only the lower log2(bitsize)
# bits of the second source.  These replacements must correctly handle the
# case where (b % bitsize) + (c % bitsize) >= bitsize.
for s in [8, 16, 32, 64]:
   mask = (1 << s) - 1

   ishl = "ishl@{}".format(s)
   ishr = "ishr@{}".format(s)
   ushr = "ushr@{}".format(s)

   in_bounds = ('ult', ('iadd', ('iand', b, mask), ('iand', c, mask)), s)

   optimizations.extend([
       ((ishl, (ishl, a, '#b'), '#c'), ('bcsel', in_bounds, (ishl, a, ('iadd', b, c)), 0)),
       ((ushr, (ushr, a, '#b'), '#c'), ('bcsel', in_bounds, (ushr, a, ('iadd', b, c)), 0)),

       # To get get -1 for large shifts of negative values, ishr must instead
       # clamp the shift count to the maximum value.
       ((ishr, (ishr, a, '#b'), '#c'),
        (ishr, a, ('imin', ('iadd', ('iand', b, mask), ('iand', c, mask)), s - 1))),
   ])

# Optimize a pattern of address calculation created by DXVK where the offset is
# divided by 4 and then multipled by 4. This can be turned into an iand and the
# additions before can be reassociated to CSE the iand instruction.
for log2 in range(1, 7): # powers of two from 2 to 64
   v = 1 << log2
   mask = 0xffffffff & ~(v - 1)
   b_is_multiple = '#b(is_unsigned_multiple_of_{})'.format(v)

   optimizations.extend([
       # 'a >> #b << #b' -> 'a & ~((1 << #b) - 1)'
       (('ishl@32', ('ushr@32', a, log2), log2), ('iand', a, mask)),

       # Reassociate for improved CSE
       (('iand@32', ('iadd@32', a, b_is_multiple), mask), ('iadd', ('iand', a, mask), b)),
   ])

# To save space in the state tables, reduce to the set that is known to help.
# Previously, this was range(1, 32).  In addition, a couple rules inside the
# loop are commented out.  Revisit someday, probably after mesa/#2635 has some
# resolution.
for i in [1, 2, 16, 24]:
    lo_mask = 0xffffffff >> i
    hi_mask = (0xffffffff << i) & 0xffffffff

    optimizations.extend([
        # This pattern seems to only help in the soft-fp64 code.
        (('ishl@32', ('iand', 'a@32', lo_mask), i), ('ishl', a, i)),
#        (('ushr@32', ('iand', 'a@32', hi_mask), i), ('ushr', a, i)),
#        (('ishr@32', ('iand', 'a@32', hi_mask), i), ('ishr', a, i)),

        (('iand', ('ishl', 'a@32', i), hi_mask), ('ishl', a, i)),
        (('iand', ('ushr', 'a@32', i), lo_mask), ('ushr', a, i)),
#        (('iand', ('ishr', 'a@32', i), lo_mask), ('ushr', a, i)), # Yes, ushr is correct
    ])

optimizations.extend([
   # This is common for address calculations.  Reassociating may enable the
   # 'a<<c' to be CSE'd.  It also helps architectures that have an ISHLADD
   # instruction or a constant offset field for in load / store instructions.
   (('ishl', ('iadd', a, '#b'), '#c'), ('iadd', ('ishl', a, c), ('ishl', b, c))),

   # Comparison simplifications
   (('~inot', ('flt', a, b)), ('fge', a, b)),
   (('~inot', ('fge', a, b)), ('flt', a, b)),
   (('inot', ('feq', a, b)), ('fne', a, b)),
   (('inot', ('fne', a, b)), ('feq', a, b)),
   (('inot', ('ilt', a, b)), ('ige', a, b)),
   (('inot', ('ult', a, b)), ('uge', a, b)),
   (('inot', ('ige', a, b)), ('ilt', a, b)),
   (('inot', ('uge', a, b)), ('ult', a, b)),
   (('inot', ('ieq', a, b)), ('ine', a, b)),
   (('inot', ('ine', a, b)), ('ieq', a, b)),

   (('iand', ('feq', a, b), ('fne', a, b)), False),
   (('iand', ('flt', a, b), ('flt', b, a)), False),
   (('iand', ('ieq', a, b), ('ine', a, b)), False),
   (('iand', ('ilt', a, b), ('ilt', b, a)), False),
   (('iand', ('ult', a, b), ('ult', b, a)), False),

   # This helps some shaders because, after some optimizations, they end up
   # with patterns like (-a < -b) || (b < a).  In an ideal world, this sort of
   # matching would be handled by CSE.
   (('flt', ('fneg', a), ('fneg', b)), ('flt', b, a)),
   (('fge', ('fneg', a), ('fneg', b)), ('fge', b, a)),
   (('feq', ('fneg', a), ('fneg', b)), ('feq', b, a)),
   (('fne', ('fneg', a), ('fneg', b)), ('fne', b, a)),
   (('flt', ('fneg', a), -1.0), ('flt', 1.0, a)),
   (('flt', -1.0, ('fneg', a)), ('flt', a, 1.0)),
   (('fge', ('fneg', a), -1.0), ('fge', 1.0, a)),
   (('fge', -1.0, ('fneg', a)), ('fge', a, 1.0)),
   (('fne', ('fneg', a), -1.0), ('fne', 1.0, a)),
   (('feq', -1.0, ('fneg', a)), ('feq', a, 1.0)),

   (('flt', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('flt', a, b)),
   (('flt', '#b(is_gt_0_and_lt_1)', ('fsat(is_used_once)', a)), ('flt', b, a)),
   (('fge', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('fge', a, b)),
   (('fge', '#b(is_gt_0_and_lt_1)', ('fsat(is_used_once)', a)), ('fge', b, a)),
   (('feq', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('feq', a, b)),
   (('fne', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('fne', a, b)),

   (('fge', ('fsat(is_used_once)', a), 1.0), ('fge', a, 1.0)),
   (('flt', ('fsat(is_used_once)', a), 1.0), ('flt', a, 1.0)),
   (('fge', 0.0, ('fsat(is_used_once)', a)), ('fge', 0.0, a)),
   (('flt', 0.0, ('fsat(is_used_once)', a)), ('flt', 0.0, a)),

   # 0.0 >= b2f(a)
   # b2f(a) <= 0.0
   # b2f(a) == 0.0 because b2f(a) can only be 0 or 1
   # inot(a)
   (('fge', 0.0, ('b2f', 'a@1')), ('inot', a)),

   (('fge', ('fneg', ('b2f', 'a@1')), 0.0), ('inot', a)),

   (('fne', ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('ior', a, b)),
   (('fne', ('fmax', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('ior', a, b)),
   (('fne', ('bcsel', a, 1.0, ('b2f', 'b@1'))   , 0.0), ('ior', a, b)),
   (('fne', ('b2f', 'a@1'), ('fneg', ('b2f', 'b@1'))),      ('ior', a, b)),
   (('fne', ('fmul', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('iand', a, b)),
   (('fne', ('fmin', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('iand', a, b)),
   (('fne', ('bcsel', a, ('b2f', 'b@1'), 0.0)   , 0.0), ('iand', a, b)),
   (('fne', ('fadd', ('b2f', 'a@1'), ('fneg', ('b2f', 'b@1'))), 0.0), ('ixor', a, b)),
   (('fne',          ('b2f', 'a@1') ,          ('b2f', 'b@1') ),      ('ixor', a, b)),
   (('fne', ('fneg', ('b2f', 'a@1')), ('fneg', ('b2f', 'b@1'))),      ('ixor', a, b)),
   (('feq', ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('inot', ('ior', a, b))),
   (('feq', ('fmax', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('inot', ('ior', a, b))),
   (('feq', ('bcsel', a, 1.0, ('b2f', 'b@1'))   , 0.0), ('inot', ('ior', a, b))),
   (('feq', ('b2f', 'a@1'), ('fneg', ('b2f', 'b@1'))),      ('inot', ('ior', a, b))),
   (('feq', ('fmul', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('inot', ('iand', a, b))),
   (('feq', ('fmin', ('b2f', 'a@1'), ('b2f', 'b@1')), 0.0), ('inot', ('iand', a, b))),
   (('feq', ('bcsel', a, ('b2f', 'b@1'), 0.0)   , 0.0), ('inot', ('iand', a, b))),
   (('feq', ('fadd', ('b2f', 'a@1'), ('fneg', ('b2f', 'b@1'))), 0.0), ('ieq', a, b)),
   (('feq',          ('b2f', 'a@1') ,          ('b2f', 'b@1') ),      ('ieq', a, b)),
   (('feq', ('fneg', ('b2f', 'a@1')), ('fneg', ('b2f', 'b@1'))),      ('ieq', a, b)),

   # -(b2f(a) + b2f(b)) < 0
   # 0 < b2f(a) + b2f(b)
   # 0 != b2f(a) + b2f(b)       b2f must be 0 or 1, so the sum is non-negative
   # a || b
   (('flt', ('fneg', ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1'))), 0.0), ('ior', a, b)),
   (('flt', 0.0, ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1'))), ('ior', a, b)),

   # -(b2f(a) + b2f(b)) >= 0
   # 0 >= b2f(a) + b2f(b)
   # 0 == b2f(a) + b2f(b)       b2f must be 0 or 1, so the sum is non-negative
   # !(a || b)
   (('fge', ('fneg', ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1'))), 0.0), ('inot', ('ior', a, b))),
   (('fge', 0.0, ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1'))), ('inot', ('ior', a, b))),

   (('flt', a, ('fneg', a)), ('flt', a, 0.0)),
   (('fge', a, ('fneg', a)), ('fge', a, 0.0)),

   # Some optimizations (below) convert things like (a < b || c < b) into
   # (min(a, c) < b).  However, this interfers with the previous optimizations
   # that try to remove comparisons with negated sums of b2f.  This just
   # breaks that apart.
   (('flt', ('fmin', c, ('fneg', ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1')))), 0.0),
    ('ior', ('flt', c, 0.0), ('ior', a, b))),

   (('~flt', ('fadd', a, b), a), ('flt', b, 0.0)),
   (('~fge', ('fadd', a, b), a), ('fge', b, 0.0)),
   (('~feq', ('fadd', a, b), a), ('feq', b, 0.0)),
   (('~fne', ('fadd', a, b), a), ('fne', b, 0.0)),
   (('~flt',                        ('fadd(is_used_once)', a, '#b'),  '#c'), ('flt', a, ('fadd', c, ('fneg', b)))),
   (('~flt', ('fneg(is_used_once)', ('fadd(is_used_once)', a, '#b')), '#c'), ('flt', ('fneg', ('fadd', c, b)), a)),
   (('~fge',                        ('fadd(is_used_once)', a, '#b'),  '#c'), ('fge', a, ('fadd', c, ('fneg', b)))),
   (('~fge', ('fneg(is_used_once)', ('fadd(is_used_once)', a, '#b')), '#c'), ('fge', ('fneg', ('fadd', c, b)), a)),
   (('~feq',                        ('fadd(is_used_once)', a, '#b'),  '#c'), ('feq', a, ('fadd', c, ('fneg', b)))),
   (('~feq', ('fneg(is_used_once)', ('fadd(is_used_once)', a, '#b')), '#c'), ('feq', ('fneg', ('fadd', c, b)), a)),
   (('~fne',                        ('fadd(is_used_once)', a, '#b'),  '#c'), ('fne', a, ('fadd', c, ('fneg', b)))),
   (('~fne', ('fneg(is_used_once)', ('fadd(is_used_once)', a, '#b')), '#c'), ('fne', ('fneg', ('fadd', c, b)), a)),

   # Cannot remove the addition from ilt or ige due to overflow.
   (('ieq', ('iadd', a, b), a), ('ieq', b, 0)),
   (('ine', ('iadd', a, b), a), ('ine', b, 0)),

   # fmin(-b2f(a), b) >= 0.0
   # -b2f(a) >= 0.0 && b >= 0.0
   # -b2f(a) == 0.0 && b >= 0.0    -b2f can only be 0 or -1, never >0
   # b2f(a) == 0.0 && b >= 0.0
   # a == False && b >= 0.0
   # !a && b >= 0.0
   #
   # The fge in the second replacement is not a typo.  I leave the proof that
   # "fmin(-b2f(a), b) >= 0 <=> fmin(-b2f(a), b) == 0" as an exercise for the
   # reader.
   (('fge', ('fmin', ('fneg', ('b2f', 'a@1')), 'b@1'), 0.0), ('iand', ('inot', a), ('fge', b, 0.0))),
   (('feq', ('fmin', ('fneg', ('b2f', 'a@1')), 'b@1'), 0.0), ('iand', ('inot', a), ('fge', b, 0.0))),

   (('feq', ('b2f', 'a@1'), 0.0), ('inot', a)),
   (('~fne', ('b2f', 'a@1'), 0.0), a),
   (('ieq', ('b2i', 'a@1'), 0),   ('inot', a)),
   (('ine', ('b2i', 'a@1'), 0),   a),

   (('fne', ('u2f', a), 0.0), ('ine', a, 0)),
   (('feq', ('u2f', a), 0.0), ('ieq', a, 0)),
   (('fge', ('u2f', a), 0.0), True),
   (('fge', 0.0, ('u2f', a)), ('uge', 0, a)),    # ieq instead?
   (('flt', ('u2f', a), 0.0), False),
   (('flt', 0.0, ('u2f', a)), ('ult', 0, a)),    # ine instead?
   (('fne', ('i2f', a), 0.0), ('ine', a, 0)),
   (('feq', ('i2f', a), 0.0), ('ieq', a, 0)),
   (('fge', ('i2f', a), 0.0), ('ige', a, 0)),
   (('fge', 0.0, ('i2f', a)), ('ige', 0, a)),
   (('flt', ('i2f', a), 0.0), ('ilt', a, 0)),
   (('flt', 0.0, ('i2f', a)), ('ilt', 0, a)),

   # 0.0 < fabs(a)
   # fabs(a) > 0.0
   # fabs(a) != 0.0 because fabs(a) must be >= 0
   # a != 0.0
   (('~flt', 0.0, ('fabs', a)), ('fne', a, 0.0)),

   # -fabs(a) < 0.0
   # fabs(a) > 0.0
   (('~flt', ('fneg', ('fabs', a)), 0.0), ('fne', a, 0.0)),

   # 0.0 >= fabs(a)
   # 0.0 == fabs(a)   because fabs(a) must be >= 0
   # 0.0 == a
   (('fge', 0.0, ('fabs', a)), ('feq', a, 0.0)),

   # -fabs(a) >= 0.0
   # 0.0 >= fabs(a)
   (('fge', ('fneg', ('fabs', a)), 0.0), ('feq', a, 0.0)),

   # (a >= 0.0) && (a <= 1.0) -> fsat(a) == a
   (('iand', ('fge', a, 0.0), ('fge', 1.0, a)), ('feq', a, ('fsat', a)), '!options->lower_fsat'),

   # (a < 0.0) || (a > 1.0)
   # !(!(a < 0.0) && !(a > 1.0))
   # !((a >= 0.0) && (a <= 1.0))
   # !(a == fsat(a))
   # a != fsat(a)
   (('ior', ('flt', a, 0.0), ('flt', 1.0, a)), ('fne', a, ('fsat', a)), '!options->lower_fsat'),

   (('fmax',                        ('b2f(is_used_once)', 'a@1'),           ('b2f', 'b@1')),           ('b2f', ('ior', a, b))),
   (('fmax', ('fneg(is_used_once)', ('b2f(is_used_once)', 'a@1')), ('fneg', ('b2f', 'b@1'))), ('fneg', ('b2f', ('ior', a, b)))),
   (('fmin',                        ('b2f(is_used_once)', 'a@1'),           ('b2f', 'b@1')),           ('b2f', ('iand', a, b))),
   (('fmin', ('fneg(is_used_once)', ('b2f(is_used_once)', 'a@1')), ('fneg', ('b2f', 'b@1'))), ('fneg', ('b2f', ('iand', a, b)))),

   # fmin(b2f(a), b)
   # bcsel(a, fmin(b2f(a), b), fmin(b2f(a), b))
   # bcsel(a, fmin(b2f(True), b), fmin(b2f(False), b))
   # bcsel(a, fmin(1.0, b), fmin(0.0, b))
   #
   # Since b is a constant, constant folding will eliminate the fmin and the
   # fmax.  If b is > 1.0, the bcsel will be replaced with a b2f.
   (('fmin', ('b2f', 'a@1'), '#b'), ('bcsel', a, ('fmin', b, 1.0), ('fmin', b, 0.0))),

   (('flt', ('fadd(is_used_once)', a, ('fneg', b)), 0.0), ('flt', a, b)),

   (('fge', ('fneg', ('fabs', a)), 0.0), ('feq', a, 0.0)),
   (('~bcsel', ('flt', b, a), b, a), ('fmin', a, b)),
   (('~bcsel', ('flt', a, b), b, a), ('fmax', a, b)),
   (('~bcsel', ('fge', a, b), b, a), ('fmin', a, b)),
   (('~bcsel', ('fge', b, a), b, a), ('fmax', a, b)),
   (('bcsel', ('i2b', a), b, c), ('bcsel', ('ine', a, 0), b, c)),
   (('bcsel', ('inot', a), b, c), ('bcsel', a, c, b)),
   (('bcsel', a, ('bcsel', a, b, c), d), ('bcsel', a, b, d)),
   (('bcsel', a, b, ('bcsel', a, c, d)), ('bcsel', a, b, d)),
   (('bcsel', a, ('bcsel', b, c, d), ('bcsel(is_used_once)', b, c, 'e')), ('bcsel', b, c, ('bcsel', a, d, 'e'))),
   (('bcsel', a, ('bcsel(is_used_once)', b, c, d), ('bcsel', b, c, 'e')), ('bcsel', b, c, ('bcsel', a, d, 'e'))),
   (('bcsel', a, ('bcsel', b, c, d), ('bcsel(is_used_once)', b, 'e', d)), ('bcsel', b, ('bcsel', a, c, 'e'), d)),
   (('bcsel', a, ('bcsel(is_used_once)', b, c, d), ('bcsel', b, 'e', d)), ('bcsel', b, ('bcsel', a, c, 'e'), d)),
   (('bcsel', a, True, b), ('ior', a, b)),
   (('bcsel', a, a, b), ('ior', a, b)),
   (('bcsel', a, b, False), ('iand', a, b)),
   (('bcsel', a, b, a), ('iand', a, b)),
   (('~fmin', a, a), a),
   (('~fmax', a, a), a),
   (('imin', a, a), a),
   (('imax', a, a), a),
   (('umin', a, a), a),
   (('umax', a, a), a),
   (('fmax', ('fmax', a, b), b), ('fmax', a, b)),
   (('umax', ('umax', a, b), b), ('umax', a, b)),
   (('imax', ('imax', a, b), b), ('imax', a, b)),
   (('fmin', ('fmin', a, b), b), ('fmin', a, b)),
   (('umin', ('umin', a, b), b), ('umin', a, b)),
   (('imin', ('imin', a, b), b), ('imin', a, b)),
   (('iand@32', a, ('inot', ('ishr', a, 31))), ('imax', a, 0)),

   # Simplify logic to detect sign of an integer.
   (('ieq', ('iand', a, 0x80000000), 0x00000000), ('ige', a, 0)),
   (('ine', ('iand', a, 0x80000000), 0x80000000), ('ige', a, 0)),
   (('ine', ('iand', a, 0x80000000), 0x00000000), ('ilt', a, 0)),
   (('ieq', ('iand', a, 0x80000000), 0x80000000), ('ilt', a, 0)),
   (('ine', ('ushr', 'a@32', 31), 0), ('ilt', a, 0)),
   (('ieq', ('ushr', 'a@32', 31), 0), ('ige', a, 0)),
   (('ieq', ('ushr', 'a@32', 31), 1), ('ilt', a, 0)),
   (('ine', ('ushr', 'a@32', 31), 1), ('ige', a, 0)),
   (('ine', ('ishr', 'a@32', 31), 0), ('ilt', a, 0)),
   (('ieq', ('ishr', 'a@32', 31), 0), ('ige', a, 0)),
   (('ieq', ('ishr', 'a@32', 31), -1), ('ilt', a, 0)),
   (('ine', ('ishr', 'a@32', 31), -1), ('ige', a, 0)),

   (('fmin', a, ('fneg', a)), ('fneg', ('fabs', a))),
   (('imin', a, ('ineg', a)), ('ineg', ('iabs', a))),
   (('fmin', a, ('fneg', ('fabs', a))), ('fneg', ('fabs', a))),
   (('imin', a, ('ineg', ('iabs', a))), ('ineg', ('iabs', a))),
   (('~fmin', a, ('fabs', a)), a),
   (('imin', a, ('iabs', a)), a),
   (('~fmax', a, ('fneg', ('fabs', a))), a),
   (('imax', a, ('ineg', ('iabs', a))), a),
   (('fmax', a, ('fabs', a)), ('fabs', a)),
   (('imax', a, ('iabs', a)), ('iabs', a)),
   (('fmax', a, ('fneg', a)), ('fabs', a)),
   (('imax', a, ('ineg', a)), ('iabs', a)),
   (('~fmax', ('fabs', a), 0.0), ('fabs', a)),
   (('~fmin', ('fmax', a, 0.0), 1.0), ('fsat', a), '!options->lower_fsat'),
   (('~fmax', ('fmin', a, 1.0), 0.0), ('fsat', a), '!options->lower_fsat'),
   (('~fmin', ('fmax', a, -1.0),  0.0), ('fneg', ('fsat', ('fneg', a))), '!options->lower_fsat'),
   (('~fmax', ('fmin', a,  0.0), -1.0), ('fneg', ('fsat', ('fneg', a))), '!options->lower_fsat'),
   (('fsat', ('fsign', a)), ('b2f', ('flt', 0.0, a))),
   (('fsat', ('b2f', a)), ('b2f', a)),
   (('fsat', a), ('fmin', ('fmax', a, 0.0), 1.0), 'options->lower_fsat'),
   (('fsat', ('fsat', a)), ('fsat', a)),
   (('fsat', ('fneg(is_used_once)', ('fadd(is_used_once)', a, b))), ('fsat', ('fadd', ('fneg', a), ('fneg', b))), '!options->lower_fsat'),
   (('fsat', ('fneg(is_used_once)', ('fmul(is_used_once)', a, b))), ('fsat', ('fmul', ('fneg', a), b)), '!options->lower_fsat'),
   (('fsat', ('fabs(is_used_once)', ('fmul(is_used_once)', a, b))), ('fsat', ('fmul', ('fabs', a), ('fabs', b))), '!options->lower_fsat'),
   (('fmin', ('fmax', ('fmin', ('fmax', a, b), c), b), c), ('fmin', ('fmax', a, b), c)),
   (('imin', ('imax', ('imin', ('imax', a, b), c), b), c), ('imin', ('imax', a, b), c)),
   (('umin', ('umax', ('umin', ('umax', a, b), c), b), c), ('umin', ('umax', a, b), c)),
   (('fmax', ('fsat', a), '#b@32(is_zero_to_one)'), ('fsat', ('fmax', a, b))),
   (('fmin', ('fsat', a), '#b@32(is_zero_to_one)'), ('fsat', ('fmin', a, b))),

   # If a in [0,b] then b-a is also in [0,b].  Since b in [0,1], max(b-a, 0) =
   # fsat(b-a).
   #
   # If a > b, then b-a < 0 and max(b-a, 0) = fsat(b-a) = 0
   #
   # This should be NaN safe since max(NaN, 0) = fsat(NaN) = 0.
   (('fmax', ('fadd(is_used_once)', ('fneg', 'a(is_not_negative)'), '#b@32(is_zero_to_one)'), 0.0),
    ('fsat', ('fadd', ('fneg',  a), b)), '!options->lower_fsat'),

   (('extract_u8', ('imin', ('imax', a, 0), 0xff), 0), ('imin', ('imax', a, 0), 0xff)),
   (('~ior', ('flt(is_used_once)', a, b), ('flt', a, c)), ('flt', a, ('fmax', b, c))),
   (('~ior', ('flt(is_used_once)', a, c), ('flt', b, c)), ('flt', ('fmin', a, b), c)),
   (('~ior', ('fge(is_used_once)', a, b), ('fge', a, c)), ('fge', a, ('fmin', b, c))),
   (('~ior', ('fge(is_used_once)', a, c), ('fge', b, c)), ('fge', ('fmax', a, b), c)),
   (('~ior', ('flt', a, '#b'), ('flt', a, '#c')), ('flt', a, ('fmax', b, c))),
   (('~ior', ('flt', '#a', c), ('flt', '#b', c)), ('flt', ('fmin', a, b), c)),
   (('~ior', ('fge', a, '#b'), ('fge', a, '#c')), ('fge', a, ('fmin', b, c))),
   (('~ior', ('fge', '#a', c), ('fge', '#b', c)), ('fge', ('fmax', a, b), c)),
   (('~iand', ('flt(is_used_once)', a, b), ('flt', a, c)), ('flt', a, ('fmin', b, c))),
   (('~iand', ('flt(is_used_once)', a, c), ('flt', b, c)), ('flt', ('fmax', a, b), c)),
   (('~iand', ('fge(is_used_once)', a, b), ('fge', a, c)), ('fge', a, ('fmax', b, c))),
   (('~iand', ('fge(is_used_once)', a, c), ('fge', b, c)), ('fge', ('fmin', a, b), c)),
   (('~iand', ('flt', a, '#b'), ('flt', a, '#c')), ('flt', a, ('fmin', b, c))),
   (('~iand', ('flt', '#a', c), ('flt', '#b', c)), ('flt', ('fmax', a, b), c)),
   (('~iand', ('fge', a, '#b'), ('fge', a, '#c')), ('fge', a, ('fmax', b, c))),
   (('~iand', ('fge', '#a', c), ('fge', '#b', c)), ('fge', ('fmin', a, b), c)),

   (('ior', ('ilt(is_used_once)', a, b), ('ilt', a, c)), ('ilt', a, ('imax', b, c))),
   (('ior', ('ilt(is_used_once)', a, c), ('ilt', b, c)), ('ilt', ('imin', a, b), c)),
   (('ior', ('ige(is_used_once)', a, b), ('ige', a, c)), ('ige', a, ('imin', b, c))),
   (('ior', ('ige(is_used_once)', a, c), ('ige', b, c)), ('ige', ('imax', a, b), c)),
   (('ior', ('ult(is_used_once)', a, b), ('ult', a, c)), ('ult', a, ('umax', b, c))),
   (('ior', ('ult(is_used_once)', a, c), ('ult', b, c)), ('ult', ('umin', a, b), c)),
   (('ior', ('uge(is_used_once)', a, b), ('uge', a, c)), ('uge', a, ('umin', b, c))),
   (('ior', ('uge(is_used_once)', a, c), ('uge', b, c)), ('uge', ('umax', a, b), c)),
   (('iand', ('ilt(is_used_once)', a, b), ('ilt', a, c)), ('ilt', a, ('imin', b, c))),
   (('iand', ('ilt(is_used_once)', a, c), ('ilt', b, c)), ('ilt', ('imax', a, b), c)),
   (('iand', ('ige(is_used_once)', a, b), ('ige', a, c)), ('ige', a, ('imax', b, c))),
   (('iand', ('ige(is_used_once)', a, c), ('ige', b, c)), ('ige', ('imin', a, b), c)),
   (('iand', ('ult(is_used_once)', a, b), ('ult', a, c)), ('ult', a, ('umin', b, c))),
   (('iand', ('ult(is_used_once)', a, c), ('ult', b, c)), ('ult', ('umax', a, b), c)),
   (('iand', ('uge(is_used_once)', a, b), ('uge', a, c)), ('uge', a, ('umax', b, c))),
   (('iand', ('uge(is_used_once)', a, c), ('uge', b, c)), ('uge', ('umin', a, b), c)),

   # These derive from the previous patterns with the application of b < 0 <=>
   # 0 < -b.  The transformation should be applied if either comparison is
   # used once as this ensures that the number of comparisons will not
   # increase.  The sources to the ior and iand are not symmetric, so the
   # rules have to be duplicated to get this behavior.
   (('~ior', ('flt(is_used_once)', 0.0, 'a@32'), ('flt', 'b@32', 0.0)), ('flt', 0.0, ('fmax', a, ('fneg', b)))),
   (('~ior', ('flt', 0.0, 'a@32'), ('flt(is_used_once)', 'b@32', 0.0)), ('flt', 0.0, ('fmax', a, ('fneg', b)))),
   (('~ior', ('fge(is_used_once)', 0.0, 'a@32'), ('fge', 'b@32', 0.0)), ('fge', 0.0, ('fmin', a, ('fneg', b)))),
   (('~ior', ('fge', 0.0, 'a@32'), ('fge(is_used_once)', 'b@32', 0.0)), ('fge', 0.0, ('fmin', a, ('fneg', b)))),
   (('~iand', ('flt(is_used_once)', 0.0, 'a@32'), ('flt', 'b@32', 0.0)), ('flt', 0.0, ('fmin', a, ('fneg', b)))),
   (('~iand', ('flt', 0.0, 'a@32'), ('flt(is_used_once)', 'b@32', 0.0)), ('flt', 0.0, ('fmin', a, ('fneg', b)))),
   (('~iand', ('fge(is_used_once)', 0.0, 'a@32'), ('fge', 'b@32', 0.0)), ('fge', 0.0, ('fmax', a, ('fneg', b)))),
   (('~iand', ('fge', 0.0, 'a@32'), ('fge(is_used_once)', 'b@32', 0.0)), ('fge', 0.0, ('fmax', a, ('fneg', b)))),

   # Common pattern like 'if (i == 0 || i == 1 || ...)'
   (('ior', ('ieq', a, 0), ('ieq', a, 1)), ('uge', 1, a)),
   (('ior', ('uge', 1, a), ('ieq', a, 2)), ('uge', 2, a)),
   (('ior', ('uge', 2, a), ('ieq', a, 3)), ('uge', 3, a)),

   # The (i2f32, ...) part is an open-coded fsign.  When that is combined with
   # the bcsel, it's basically copysign(1.0, a).  There is no copysign in NIR,
   # so emit an open-coded version of that.
   (('bcsel@32', ('feq', a, 0.0), 1.0, ('i2f32', ('iadd', ('b2i32', ('flt', 0.0, 'a@32')), ('ineg', ('b2i32', ('flt', 'a@32', 0.0)))))),
    ('ior', 0x3f800000, ('iand', a, 0x80000000))),

   (('ior', a, ('ieq', a, False)), True),
   (('ior', a, ('inot', a)), -1),

   (('ine', ('ineg', ('b2i32', 'a@1')), ('ineg', ('b2i32', 'b@1'))), ('ine', a, b)),
   (('b2i32', ('ine', 'a@1', 'b@1')), ('b2i32', ('ixor', a, b))),

   (('iand', ('ieq', 'a@32', 0), ('ieq', 'b@32', 0)), ('ieq', ('ior', a, b), 0), '!options->lower_bitops'),
   (('ior',  ('ine', 'a@32', 0), ('ine', 'b@32', 0)), ('ine', ('ior', a, b), 0), '!options->lower_bitops'),

   # This pattern occurs coutresy of __flt64_nonnan in the soft-fp64 code.
   # The first part of the iand comes from the !__feq64_nonnan.
   #
   # The second pattern is a reformulation of the first based on the relation
   # (a == 0 || y == 0) <=> umin(a, y) == 0, where b in the first equation
   # happens to be y == 0.
   (('iand', ('inot', ('iand', ('ior', ('ieq', a, 0),  b), c)), ('ilt', a, 0)),
    ('iand', ('inot', ('iand',                         b , c)), ('ilt', a, 0))),
   (('iand', ('inot', ('iand', ('ieq', ('umin', a, b), 0), c)), ('ilt', a, 0)),
    ('iand', ('inot', ('iand', ('ieq',             b , 0), c)), ('ilt', a, 0))),

   # These patterns can result when (a < b || a < c) => (a < min(b, c))
   # transformations occur before constant propagation and loop-unrolling.
   (('~flt', a, ('fmax', b, a)), ('flt', a, b)),
   (('~flt', ('fmin', a, b), a), ('flt', b, a)),
   (('~fge', a, ('fmin', b, a)), True),
   (('~fge', ('fmax', a, b), a), True),
   (('~flt', a, ('fmin', b, a)), False),
   (('~flt', ('fmax', a, b), a), False),
   (('~fge', a, ('fmax', b, a)), ('fge', a, b)),
   (('~fge', ('fmin', a, b), a), ('fge', b, a)),

   (('ilt', a, ('imax', b, a)), ('ilt', a, b)),
   (('ilt', ('imin', a, b), a), ('ilt', b, a)),
   (('ige', a, ('imin', b, a)), True),
   (('ige', ('imax', a, b), a), True),
   (('ult', a, ('umax', b, a)), ('ult', a, b)),
   (('ult', ('umin', a, b), a), ('ult', b, a)),
   (('uge', a, ('umin', b, a)), True),
   (('uge', ('umax', a, b), a), True),
   (('ilt', a, ('imin', b, a)), False),
   (('ilt', ('imax', a, b), a), False),
   (('ige', a, ('imax', b, a)), ('ige', a, b)),
   (('ige', ('imin', a, b), a), ('ige', b, a)),
   (('ult', a, ('umin', b, a)), False),
   (('ult', ('umax', a, b), a), False),
   (('uge', a, ('umax', b, a)), ('uge', a, b)),
   (('uge', ('umin', a, b), a), ('uge', b, a)),
   (('ult', a, ('iand', b, a)), False),
   (('ult', ('ior', a, b), a), False),
   (('uge', a, ('iand', b, a)), True),
   (('uge', ('ior', a, b), a), True),

   (('ilt', '#a', ('imax', '#b', c)), ('ior', ('ilt', a, b), ('ilt', a, c))),
   (('ilt', ('imin', '#a', b), '#c'), ('ior', ('ilt', a, c), ('ilt', b, c))),
   (('ige', '#a', ('imin', '#b', c)), ('ior', ('ige', a, b), ('ige', a, c))),
   (('ige', ('imax', '#a', b), '#c'), ('ior', ('ige', a, c), ('ige', b, c))),
   (('ult', '#a', ('umax', '#b', c)), ('ior', ('ult', a, b), ('ult', a, c))),
   (('ult', ('umin', '#a', b), '#c'), ('ior', ('ult', a, c), ('ult', b, c))),
   (('uge', '#a', ('umin', '#b', c)), ('ior', ('uge', a, b), ('uge', a, c))),
   (('uge', ('umax', '#a', b), '#c'), ('ior', ('uge', a, c), ('uge', b, c))),
   (('ilt', '#a', ('imin', '#b', c)), ('iand', ('ilt', a, b), ('ilt', a, c))),
   (('ilt', ('imax', '#a', b), '#c'), ('iand', ('ilt', a, c), ('ilt', b, c))),
   (('ige', '#a', ('imax', '#b', c)), ('iand', ('ige', a, b), ('ige', a, c))),
   (('ige', ('imin', '#a', b), '#c'), ('iand', ('ige', a, c), ('ige', b, c))),
   (('ult', '#a', ('umin', '#b', c)), ('iand', ('ult', a, b), ('ult', a, c))),
   (('ult', ('umax', '#a', b), '#c'), ('iand', ('ult', a, c), ('ult', b, c))),
   (('uge', '#a', ('umax', '#b', c)), ('iand', ('uge', a, b), ('uge', a, c))),
   (('uge', ('umin', '#a', b), '#c'), ('iand', ('uge', a, c), ('uge', b, c))),

   # Thanks to sign extension, the ishr(a, b) is negative if and only if a is
   # negative.
   (('bcsel', ('ilt', a, 0), ('ineg', ('ishr', a, b)), ('ishr', a, b)),
    ('iabs', ('ishr', a, b))),
   (('iabs', ('ishr', ('iabs', a), b)), ('ishr', ('iabs', a), b)),

   (('fabs', ('slt', a, b)), ('slt', a, b)),
   (('fabs', ('sge', a, b)), ('sge', a, b)),
   (('fabs', ('seq', a, b)), ('seq', a, b)),
   (('fabs', ('sne', a, b)), ('sne', a, b)),
   (('slt', a, b), ('b2f', ('flt', a, b)), 'options->lower_scmp'),
   (('sge', a, b), ('b2f', ('fge', a, b)), 'options->lower_scmp'),
   (('seq', a, b), ('b2f', ('feq', a, b)), 'options->lower_scmp'),
   (('sne', a, b), ('b2f', ('fne', a, b)), 'options->lower_scmp'),
   (('seq', ('seq', a, b), 1.0), ('seq', a, b)),
   (('seq', ('sne', a, b), 1.0), ('sne', a, b)),
   (('seq', ('slt', a, b), 1.0), ('slt', a, b)),
   (('seq', ('sge', a, b), 1.0), ('sge', a, b)),
   (('sne', ('seq', a, b), 0.0), ('seq', a, b)),
   (('sne', ('sne', a, b), 0.0), ('sne', a, b)),
   (('sne', ('slt', a, b), 0.0), ('slt', a, b)),
   (('sne', ('sge', a, b), 0.0), ('sge', a, b)),
   (('seq', ('seq', a, b), 0.0), ('sne', a, b)),
   (('seq', ('sne', a, b), 0.0), ('seq', a, b)),
   (('seq', ('slt', a, b), 0.0), ('sge', a, b)),
   (('seq', ('sge', a, b), 0.0), ('slt', a, b)),
   (('sne', ('seq', a, b), 1.0), ('sne', a, b)),
   (('sne', ('sne', a, b), 1.0), ('seq', a, b)),
   (('sne', ('slt', a, b), 1.0), ('sge', a, b)),
   (('sne', ('sge', a, b), 1.0), ('slt', a, b)),
   (('fall_equal2', a, b), ('fmin', ('seq', 'a.x', 'b.x'), ('seq', 'a.y', 'b.y')), 'options->lower_vector_cmp'),
   (('fall_equal3', a, b), ('seq', ('fany_nequal3', a, b), 0.0), 'options->lower_vector_cmp'),
   (('fall_equal4', a, b), ('seq', ('fany_nequal4', a, b), 0.0), 'options->lower_vector_cmp'),
   (('fany_nequal2', a, b), ('fmax', ('sne', 'a.x', 'b.x'), ('sne', 'a.y', 'b.y')), 'options->lower_vector_cmp'),
   (('fany_nequal3', a, b), ('fsat', ('fdot3', ('sne', a, b), ('sne', a, b))), 'options->lower_vector_cmp'),
   (('fany_nequal4', a, b), ('fsat', ('fdot4', ('sne', a, b), ('sne', a, b))), 'options->lower_vector_cmp'),
   (('fne', ('fneg', a), a), ('fne', a, 0.0)),
   (('feq', ('fneg', a), a), ('feq', a, 0.0)),
   # Emulating booleans
   (('imul', ('b2i', 'a@1'), ('b2i', 'b@1')), ('b2i', ('iand', a, b))),
   (('fmul', ('b2f', 'a@1'), ('b2f', 'b@1')), ('b2f', ('iand', a, b))),
   (('fsat', ('fadd', ('b2f', 'a@1'), ('b2f', 'b@1'))), ('b2f', ('ior', a, b))),
   (('iand', 'a@bool32', 1.0), ('b2f', a)),
   # True/False are ~0 and 0 in NIR.  b2i of True is 1, and -1 is ~0 (True).
   (('ineg', ('b2i32', 'a@32')), a),
   (('flt', ('fneg', ('b2f', 'a@1')), 0), a), # Generated by TGSI KILL_IF.
   # Comparison with the same args.  Note that these are not done for
   # the float versions because NaN always returns false on float
   # inequalities.
   (('ilt', a, a), False),
   (('ige', a, a), True),
   (('ieq', a, a), True),
   (('ine', a, a), False),
   (('ult', a, a), False),
   (('uge', a, a), True),
   # Logical and bit operations
   (('iand', a, a), a),
   (('iand', a, ~0), a),
   (('iand', a, 0), 0),
   (('ior', a, a), a),
   (('ior', a, 0), a),
   (('ior', a, True), True),
   (('ixor', a, a), 0),
   (('ixor', a, 0), a),
   (('inot', ('inot', a)), a),
   (('ior', ('iand', a, b), b), b),
   (('ior', ('ior', a, b), b), ('ior', a, b)),
   (('iand', ('ior', a, b), b), b),
   (('iand', ('iand', a, b), b), ('iand', a, b)),
   # DeMorgan's Laws
   (('iand', ('inot', a), ('inot', b)), ('inot', ('ior',  a, b))),
   (('ior',  ('inot', a), ('inot', b)), ('inot', ('iand', a, b))),
   # Shift optimizations
   (('ishl', 0, a), 0),
   (('ishl', a, 0), a),
   (('ishr', 0, a), 0),
   (('ishr', a, 0), a),
   (('ushr', 0, a), 0),
   (('ushr', a, 0), a),
   (('ior', ('ishl@16', a, b), ('ushr@16', a, ('iadd', 16, ('ineg', b)))), ('urol', a, b), '!options->lower_rotate'),
   (('ior', ('ishl@16', a, b), ('ushr@16', a, ('isub', 16, b))), ('urol', a, b), '!options->lower_rotate'),
   (('ior', ('ishl@32', a, b), ('ushr@32', a, ('iadd', 32, ('ineg', b)))), ('urol', a, b), '!options->lower_rotate'),
   (('ior', ('ishl@32', a, b), ('ushr@32', a, ('isub', 32, b))), ('urol', a, b), '!options->lower_rotate'),
   (('ior', ('ushr@16', a, b), ('ishl@16', a, ('iadd', 16, ('ineg', b)))), ('uror', a, b), '!options->lower_rotate'),
   (('ior', ('ushr@16', a, b), ('ishl@16', a, ('isub', 16, b))), ('uror', a, b), '!options->lower_rotate'),
   (('ior', ('ushr@32', a, b), ('ishl@32', a, ('iadd', 32, ('ineg', b)))), ('uror', a, b), '!options->lower_rotate'),
   (('ior', ('ushr@32', a, b), ('ishl@32', a, ('isub', 32, b))), ('uror', a, b), '!options->lower_rotate'),
   (('urol@16', a, b), ('ior', ('ishl', a, b), ('ushr', a, ('isub', 16, b))), 'options->lower_rotate'),
   (('urol@32', a, b), ('ior', ('ishl', a, b), ('ushr', a, ('isub', 32, b))), 'options->lower_rotate'),
   (('uror@16', a, b), ('ior', ('ushr', a, b), ('ishl', a, ('isub', 16, b))), 'options->lower_rotate'),
   (('uror@32', a, b), ('ior', ('ushr', a, b), ('ishl', a, ('isub', 32, b))), 'options->lower_rotate'),
   # Exponential/logarithmic identities
   (('~fexp2', ('flog2', a)), a), # 2^lg2(a) = a
   (('~flog2', ('fexp2', a)), a), # lg2(2^a) = a
   (('fpow', a, b), ('fexp2', ('fmul', ('flog2', a), b)), 'options->lower_fpow'), # a^b = 2^(lg2(a)*b)
   (('~fexp2', ('fmul', ('flog2', a), b)), ('fpow', a, b), '!options->lower_fpow'), # 2^(lg2(a)*b) = a^b
   (('~fexp2', ('fadd', ('fmul', ('flog2', a), b), ('fmul', ('flog2', c), d))),
    ('~fmul', ('fpow', a, b), ('fpow', c, d)), '!options->lower_fpow'), # 2^(lg2(a) * b + lg2(c) + d) = a^b * c^d
   (('~fexp2', ('fmul', ('flog2', a), 0.5)), ('fsqrt', a)),
   (('~fexp2', ('fmul', ('flog2', a), 2.0)), ('fmul', a, a)),
   (('~fexp2', ('fmul', ('flog2', a), 4.0)), ('fmul', ('fmul', a, a), ('fmul', a, a))),
   (('~fpow', a, 1.0), a),
   (('~fpow', a, 2.0), ('fmul', a, a)),
   (('~fpow', a, 4.0), ('fmul', ('fmul', a, a), ('fmul', a, a))),
   (('~fpow', 2.0, a), ('fexp2', a)),
   (('~fpow', ('fpow', a, 2.2), 0.454545), a),
   (('~fpow', ('fabs', ('fpow', a, 2.2)), 0.454545), ('fabs', a)),
   (('~fsqrt', ('fexp2', a)), ('fexp2', ('fmul', 0.5, a))),
   (('~frcp', ('fexp2', a)), ('fexp2', ('fneg', a))),
   (('~frsq', ('fexp2', a)), ('fexp2', ('fmul', -0.5, a))),
   (('~flog2', ('fsqrt', a)), ('fmul', 0.5, ('flog2', a))),
   (('~flog2', ('frcp', a)), ('fneg', ('flog2', a))),
   (('~flog2', ('frsq', a)), ('fmul', -0.5, ('flog2', a))),
   (('~flog2', ('fpow', a, b)), ('fmul', b, ('flog2', a))),
   (('~fmul', ('fexp2(is_used_once)', a), ('fexp2(is_used_once)', b)), ('fexp2', ('fadd', a, b))),
   (('bcsel', ('flt', a, 0.0), 0.0, ('fsqrt', a)), ('fsqrt', ('fmax', a, 0.0))),
   (('~fmul', ('fsqrt', a), ('fsqrt', a)), ('fabs',a)),
   # Division and reciprocal
   (('~fdiv', 1.0, a), ('frcp', a)),
   (('fdiv', a, b), ('fmul', a, ('frcp', b)), 'options->lower_fdiv'),
   (('~frcp', ('frcp', a)), a),
   (('~frcp', ('fsqrt', a)), ('frsq', a)),
   (('fsqrt', a), ('frcp', ('frsq', a)), 'options->lower_fsqrt'),
   (('~frcp', ('frsq', a)), ('fsqrt', a), '!options->lower_fsqrt'),
   # Trig
   (('fsin', a), lowered_sincos(0.5), 'options->lower_sincos'),
   (('fcos', a), lowered_sincos(0.75), 'options->lower_sincos'),
   # Boolean simplifications
   (('i2b32(is_used_by_if)', a), ('ine32', a, 0)),
   (('i2b1(is_used_by_if)', a), ('ine', a, 0)),
   (('ieq', a, True), a),
   (('ine(is_not_used_by_if)', a, True), ('inot', a)),
   (('ine', a, False), a),
   (('ieq(is_not_used_by_if)', a, False), ('inot', 'a')),
   (('bcsel', a, True, False), a),
   (('bcsel', a, False, True), ('inot', a)),
   (('bcsel@32', a, 1.0, 0.0), ('b2f', a)),
   (('bcsel@32', a, 0.0, 1.0), ('b2f', ('inot', a))),
   (('bcsel@32', a, -1.0, -0.0), ('fneg', ('b2f', a))),
   (('bcsel@32', a, -0.0, -1.0), ('fneg', ('b2f', ('inot', a)))),
   (('bcsel', True, b, c), b),
   (('bcsel', False, b, c), c),
   (('bcsel', a, ('b2f(is_used_once)', 'b@32'), ('b2f', 'c@32')), ('b2f', ('bcsel', a, b, c))),

   (('bcsel', a, b, b), b),
   (('~fcsel', a, b, b), b),

   # D3D Boolean emulation
   (('bcsel', a, -1, 0), ('ineg', ('b2i', 'a@1'))),
   (('bcsel', a, 0, -1), ('ineg', ('b2i', ('inot', a)))),
   (('iand', ('ineg', ('b2i', 'a@1')), ('ineg', ('b2i', 'b@1'))),
    ('ineg', ('b2i', ('iand', a, b)))),
   (('ior', ('ineg', ('b2i','a@1')), ('ineg', ('b2i', 'b@1'))),
    ('ineg', ('b2i', ('ior', a, b)))),
   (('ieq', ('ineg', ('b2i', 'a@1')), 0), ('inot', a)),
   (('ieq', ('ineg', ('b2i', 'a@1')), -1), a),
   (('ine', ('ineg', ('b2i', 'a@1')), 0), a),
   (('ine', ('ineg', ('b2i', 'a@1')), -1), ('inot', a)),
   (('iand', ('ineg', ('b2i', a)), 1.0), ('b2f', a)),
   (('iand', ('ineg', ('b2i', a)), 1),   ('b2i', a)),

   # SM5 32-bit shifts are defined to use the 5 least significant bits
   (('ishl', 'a@32', ('iand', 31, b)), ('ishl', a, b)),
   (('ishr', 'a@32', ('iand', 31, b)), ('ishr', a, b)),
   (('ushr', 'a@32', ('iand', 31, b)), ('ushr', a, b)),

   # Conversions
   (('i2b32', ('b2i', 'a@32')), a),
   (('f2i', ('ftrunc', a)), ('f2i', a)),
   (('f2u', ('ftrunc', a)), ('f2u', a)),
   (('i2b', ('ineg', a)), ('i2b', a)),
   (('i2b', ('iabs', a)), ('i2b', a)),
   (('inot', ('f2b1', a)), ('feq', a, 0.0)),

   # The C spec says, "If the value of the integral part cannot be represented
   # by the integer type, the behavior is undefined."  "Undefined" can mean
   # "the conversion doesn't happen at all."
   (('~i2f32', ('f2i32', 'a@32')), ('ftrunc', a)),

   # Ironically, mark these as imprecise because removing the conversions may
   # preserve more precision than doing the conversions (e.g.,
   # uint(float(0x81818181u)) == 0x81818200).
   (('~f2i32', ('i2f', 'a@32')), a),
   (('~f2i32', ('u2f', 'a@32')), a),
   (('~f2u32', ('i2f', 'a@32')), a),
   (('~f2u32', ('u2f', 'a@32')), a),

   # Conversions from float16 to float32 and back can always be removed
   (('f2f16', ('f2f32', 'a@16')), a),
   (('f2fmp', ('f2f32', 'a@16')), a),
   # Conversions to float16 would be lossy so they should only be removed if
   # the instruction was generated by the precision lowering pass.
   (('f2f32', ('f2fmp', 'a@32')), a),

   (('ffloor', 'a(is_integral)'), a),
   (('fceil', 'a(is_integral)'), a),
   (('ftrunc', 'a(is_integral)'), a),
   # fract(x) = x - floor(x), so fract(NaN) = NaN
   (('~ffract', 'a(is_integral)'), 0.0),
   (('fabs', 'a(is_not_negative)'), a),
   (('iabs', 'a(is_not_negative)'), a),
   (('fsat', 'a(is_not_positive)'), 0.0),

   # Section 5.4.1 (Conversion and Scalar Constructors) of the GLSL 4.60 spec
   # says:
   #
   #    It is undefined to convert a negative floating-point value to an
   #    uint.
   #
   # Assuming that (uint)some_float behaves like (uint)(int)some_float allows
   # some optimizations in the i965 backend to proceed.
   (('ige', ('f2u', a), b), ('ige', ('f2i', a), b)),
   (('ige', b, ('f2u', a)), ('ige', b, ('f2i', a))),
   (('ilt', ('f2u', a), b), ('ilt', ('f2i', a), b)),
   (('ilt', b, ('f2u', a)), ('ilt', b, ('f2i', a))),

   (('~fmin', 'a(is_not_negative)', 1.0), ('fsat', a), '!options->lower_fsat'),

   # The result of the multiply must be in [-1, 0], so the result of the ffma
   # must be in [0, 1].
   (('flt', ('fadd', ('fmul', ('fsat', a), ('fneg', ('fsat', a))), 1.0), 0.0), False),
   (('flt', ('fadd', ('fneg', ('fmul', ('fsat', a), ('fsat', a))), 1.0), 0.0), False),
   (('fmax', ('fadd', ('fmul', ('fsat', a), ('fneg', ('fsat', a))), 1.0), 0.0), ('fadd', ('fmul', ('fsat', a), ('fneg', ('fsat', a))), 1.0)),
   (('fmax', ('fadd', ('fneg', ('fmul', ('fsat', a), ('fsat', a))), 1.0), 0.0), ('fadd', ('fneg', ('fmul', ('fsat', a), ('fsat', a))), 1.0)),

   (('fne', 'a(is_not_zero)', 0.0), True),
   (('feq', 'a(is_not_zero)', 0.0), False),

   # In this chart, + means value > 0 and - means value < 0.
   #
   # + >= + -> unknown  0 >= + -> false    - >= + -> false
   # + >= 0 -> true     0 >= 0 -> true     - >= 0 -> false
   # + >= - -> true     0 >= - -> true     - >= - -> unknown
   #
   # Using grouping conceptually similar to a Karnaugh map...
   #
   # (+ >= 0, + >= -, 0 >= 0, 0 >= -) == (is_not_negative >= is_not_positive) -> true
   # (0 >= +, - >= +) == (is_not_positive >= gt_zero) -> false
   # (- >= +, - >= 0) == (lt_zero >= is_not_negative) -> false
   #
   # The flt / ilt cases just invert the expected result.
   #
   # The results expecting true, must be marked imprecise.  The results
   # expecting false are fine because NaN compared >= or < anything is false.

   (('~fge', 'a(is_not_negative)', 'b(is_not_positive)'), True),
   (('fge',  'a(is_not_positive)', 'b(is_gt_zero)'),      False),
   (('fge',  'a(is_lt_zero)',      'b(is_not_negative)'), False),

   (('flt',  'a(is_not_negative)', 'b(is_not_positive)'), False),
   (('~flt', 'a(is_not_positive)', 'b(is_gt_zero)'),      True),
   (('~flt', 'a(is_lt_zero)',      'b(is_not_negative)'), True),

   (('ine', 'a(is_not_zero)', 0), True),
   (('ieq', 'a(is_not_zero)', 0), False),

   (('ige', 'a(is_not_negative)', 'b(is_not_positive)'), True),
   (('ige', 'a(is_not_positive)', 'b(is_gt_zero)'),      False),
   (('ige', 'a(is_lt_zero)',      'b(is_not_negative)'), False),

   (('ilt', 'a(is_not_negative)', 'b(is_not_positive)'), False),
   (('ilt', 'a(is_not_positive)', 'b(is_gt_zero)'),      True),
   (('ilt', 'a(is_lt_zero)',      'b(is_not_negative)'), True),

   (('ult', 0, 'a(is_gt_zero)'), True),
   (('ult', a, 0), False),

   # Packing and then unpacking does nothing
   (('unpack_64_2x32_split_x', ('pack_64_2x32_split', a, b)), a),
   (('unpack_64_2x32_split_y', ('pack_64_2x32_split', a, b)), b),
   (('pack_64_2x32_split', ('unpack_64_2x32_split_x', a),
                           ('unpack_64_2x32_split_y', a)), a),

   # Comparing two halves of an unpack separately.  While this optimization
   # should be correct for non-constant values, it's less obvious that it's
   # useful in that case.  For constant values, the pack will fold and we're
   # guaranteed to reduce the whole tree to one instruction.
   (('iand', ('ieq', ('unpack_32_2x16_split_x', a), '#b'),
             ('ieq', ('unpack_32_2x16_split_y', a), '#c')),
    ('ieq', a, ('pack_32_2x16_split', b, c))),

   # Byte extraction
   (('ushr', 'a@16',  8), ('extract_u8', a, 1), '!options->lower_extract_byte'),
   (('ushr', 'a@32', 24), ('extract_u8', a, 3), '!options->lower_extract_byte'),
   (('ushr', 'a@64', 56), ('extract_u8', a, 7), '!options->lower_extract_byte'),
   (('ishr', 'a@16',  8), ('extract_i8', a, 1), '!options->lower_extract_byte'),
   (('ishr', 'a@32', 24), ('extract_i8', a, 3), '!options->lower_extract_byte'),
   (('ishr', 'a@64', 56), ('extract_i8', a, 7), '!options->lower_extract_byte'),
   (('iand', 0xff, a), ('extract_u8', a, 0), '!options->lower_extract_byte'),

   # Useless masking before unpacking
   (('unpack_half_2x16_split_x', ('iand', a, 0xffff)), ('unpack_half_2x16_split_x', a)),
   (('unpack_32_2x16_split_x', ('iand', a, 0xffff)), ('unpack_32_2x16_split_x', a)),
   (('unpack_64_2x32_split_x', ('iand', a, 0xffffffff)), ('unpack_64_2x32_split_x', a)),
   (('unpack_half_2x16_split_y', ('iand', a, 0xffff0000)), ('unpack_half_2x16_split_y', a)),
   (('unpack_32_2x16_split_y', ('iand', a, 0xffff0000)), ('unpack_32_2x16_split_y', a)),
   (('unpack_64_2x32_split_y', ('iand', a, 0xffffffff00000000)), ('unpack_64_2x32_split_y', a)),

   # Optimize half packing
   (('ishl', ('pack_half_2x16', ('vec2', a, 0)), 16), ('pack_half_2x16', ('vec2', 0, a))),
   (('ishr', ('pack_half_2x16', ('vec2', 0, a)), 16), ('pack_half_2x16', ('vec2', a, 0))),

   (('iadd', ('pack_half_2x16', ('vec2', a, 0)), ('pack_half_2x16', ('vec2', 0, b))),
    ('pack_half_2x16', ('vec2', a, b))),
   (('ior', ('pack_half_2x16', ('vec2', a, 0)), ('pack_half_2x16', ('vec2', 0, b))),
    ('pack_half_2x16', ('vec2', a, b))),
])

# After the ('extract_u8', a, 0) pattern, above, triggers, there will be
# patterns like those below.
for op in ('ushr', 'ishr'):
   optimizations.extend([(('extract_u8', (op, 'a@16',  8),     0), ('extract_u8', a, 1))])
   optimizations.extend([(('extract_u8', (op, 'a@32',  8 * i), 0), ('extract_u8', a, i)) for i in range(1, 4)])
   optimizations.extend([(('extract_u8', (op, 'a@64',  8 * i), 0), ('extract_u8', a, i)) for i in range(1, 8)])

optimizations.extend([(('extract_u8', ('extract_u16', a, 1), 0), ('extract_u8', a, 2))])

# After the ('extract_[iu]8', a, 3) patterns, above, trigger, there will be
# patterns like those below.
for op in ('extract_u8', 'extract_i8'):
   optimizations.extend([((op, ('ishl', 'a@16',      8),     1), (op, a, 0))])
   optimizations.extend([((op, ('ishl', 'a@32', 24 - 8 * i), 3), (op, a, i)) for i in range(2, -1, -1)])
   optimizations.extend([((op, ('ishl', 'a@64', 56 - 8 * i), 7), (op, a, i)) for i in range(6, -1, -1)])

optimizations.extend([
    # Word extraction
   (('ushr', ('ishl', 'a@32', 16), 16), ('extract_u16', a, 0), '!options->lower_extract_word'),
   (('ushr', 'a@32', 16), ('extract_u16', a, 1), '!options->lower_extract_word'),
   (('ishr', ('ishl', 'a@32', 16), 16), ('extract_i16', a, 0), '!options->lower_extract_word'),
   (('ishr', 'a@32', 16), ('extract_i16', a, 1), '!options->lower_extract_word'),
   (('iand', 0xffff, a), ('extract_u16', a, 0), '!options->lower_extract_word'),

   # Subtracts
   (('ussub_4x8', a, 0), a),
   (('ussub_4x8', a, ~0), 0),
   # Lower all Subtractions first - they can get recombined later
   (('fsub', a, b), ('fadd', a, ('fneg', b))),
   (('isub', a, b), ('iadd', a, ('ineg', b))),
   (('uabs_usub', a, b), ('bcsel', ('ult', a, b), ('ineg', ('isub', a, b)), ('isub', a, b))),
   # This is correct.  We don't need isub_sat because the result type is unsigned, so it cannot overflow.
   (('uabs_isub', a, b), ('bcsel', ('ilt', a, b), ('ineg', ('isub', a, b)), ('isub', a, b))),

   # Propagate negation up multiplication chains
   (('fmul(is_used_by_non_fsat)', ('fneg', a), b), ('fneg', ('fmul', a, b))),
   (('imul', ('ineg', a), b), ('ineg', ('imul', a, b))),

   # Propagate constants up multiplication chains
   (('~fmul(is_used_once)', ('fmul(is_used_once)', 'a(is_not_const)', 'b(is_not_const)'), '#c'), ('fmul', ('fmul', a, c), b)),
   (('imul(is_used_once)', ('imul(is_used_once)', 'a(is_not_const)', 'b(is_not_const)'), '#c'), ('imul', ('imul', a, c), b)),
   (('~fadd(is_used_once)', ('fadd(is_used_once)', 'a(is_not_const)', 'b(is_not_const)'), '#c'), ('fadd', ('fadd', a, c), b)),
   (('iadd(is_used_once)', ('iadd(is_used_once)', 'a(is_not_const)', 'b(is_not_const)'), '#c'), ('iadd', ('iadd', a, c), b)),

   # Reassociate constants in add/mul chains so they can be folded together.
   # For now, we mostly only handle cases where the constants are separated by
   # a single non-constant.  We could do better eventually.
   (('~fmul', '#a', ('fmul', 'b(is_not_const)', '#c')), ('fmul', ('fmul', a, c), b)),
   (('imul', '#a', ('imul', 'b(is_not_const)', '#c')), ('imul', ('imul', a, c), b)),
   (('~fadd', '#a',          ('fadd', 'b(is_not_const)', '#c')),  ('fadd', ('fadd', a,          c),           b)),
   (('~fadd', '#a', ('fneg', ('fadd', 'b(is_not_const)', '#c'))), ('fadd', ('fadd', a, ('fneg', c)), ('fneg', b))),
   (('iadd', '#a', ('iadd', 'b(is_not_const)', '#c')), ('iadd', ('iadd', a, c), b)),
   (('iand', '#a', ('iand', 'b(is_not_const)', '#c')), ('iand', ('iand', a, c), b)),
   (('ior',  '#a', ('ior',  'b(is_not_const)', '#c')), ('ior',  ('ior',  a, c), b)),
   (('ixor', '#a', ('ixor', 'b(is_not_const)', '#c')), ('ixor', ('ixor', a, c), b)),

   # Drop mul-div by the same value when there's no wrapping.
   (('idiv', ('imul(no_signed_wrap)', a, b), b), a),

   # By definition...
   (('bcsel', ('ige', ('find_lsb', a), 0), ('find_lsb', a), -1), ('find_lsb', a)),
   (('bcsel', ('ige', ('ifind_msb', a), 0), ('ifind_msb', a), -1), ('ifind_msb', a)),
   (('bcsel', ('ige', ('ufind_msb', a), 0), ('ufind_msb', a), -1), ('ufind_msb', a)),

   (('bcsel', ('ine', a, 0), ('find_lsb', a), -1), ('find_lsb', a)),
   (('bcsel', ('ine', a, 0), ('ifind_msb', a), -1), ('ifind_msb', a)),
   (('bcsel', ('ine', a, 0), ('ufind_msb', a), -1), ('ufind_msb', a)),

   (('bcsel', ('ine', a, -1), ('ifind_msb', a), -1), ('ifind_msb', a)),

   # Misc. lowering
   (('fmod', a, b), ('fsub', a, ('fmul', b, ('ffloor', ('fdiv', a, b)))), 'options->lower_fmod'),
   (('frem', a, b), ('fsub', a, ('fmul', b, ('ftrunc', ('fdiv', a, b)))), 'options->lower_fmod'),
   (('uadd_carry@32', a, b), ('b2i', ('ult', ('iadd', a, b), a)), 'options->lower_uadd_carry'),
   (('usub_borrow@32', a, b), ('b2i', ('ult', a, b)), 'options->lower_usub_borrow'),

   (('bitfield_insert', 'base', 'insert', 'offset', 'bits'),
    ('bcsel', ('ult', 31, 'bits'), 'insert',
              ('bfi', ('bfm', 'bits', 'offset'), 'insert', 'base')),
    'options->lower_bitfield_insert'),
   (('ihadd', a, b), ('iadd', ('iand', a, b), ('ishr', ('ixor', a, b), 1)), 'options->lower_hadd'),
   (('uhadd', a, b), ('iadd', ('iand', a, b), ('ushr', ('ixor', a, b), 1)), 'options->lower_hadd'),
   (('irhadd', a, b), ('isub', ('ior', a, b), ('ishr', ('ixor', a, b), 1)), 'options->lower_hadd'),
   (('urhadd', a, b), ('isub', ('ior', a, b), ('ushr', ('ixor', a, b), 1)), 'options->lower_hadd'),
   (('ihadd@64', a, b), ('iadd', ('iand', a, b), ('ishr', ('ixor', a, b), 1)), 'options->lower_hadd64 || (options->lower_int64_options & nir_lower_iadd64) != 0'),
   (('uhadd@64', a, b), ('iadd', ('iand', a, b), ('ushr', ('ixor', a, b), 1)), 'options->lower_hadd64 || (options->lower_int64_options & nir_lower_iadd64) != 0'),
   (('irhadd@64', a, b), ('isub', ('ior', a, b), ('ishr', ('ixor', a, b), 1)), 'options->lower_hadd64 || (options->lower_int64_options & nir_lower_iadd64) != 0'),
   (('urhadd@64', a, b), ('isub', ('ior', a, b), ('ushr', ('ixor', a, b), 1)), 'options->lower_hadd64 || (options->lower_int64_options & nir_lower_iadd64) != 0'),

   (('uadd_sat@64', a, b), ('bcsel', ('ult', ('iadd', a, b), a), -1, ('iadd', a, b)), 'options->lower_add_sat || (options->lower_int64_options & nir_lower_iadd64) != 0'),
   (('uadd_sat', a, b), ('bcsel', ('ult', ('iadd', a, b), a), -1, ('iadd', a, b)), 'options->lower_add_sat'),
   (('usub_sat', a, b), ('bcsel', ('ult', a, b), 0, ('isub', a, b)), 'options->lower_add_sat'),
   (('usub_sat@64', a, b), ('bcsel', ('ult', a, b), 0, ('isub', a, b)), 'options->lower_usub_sat64 || (options->lower_int64_options & nir_lower_iadd64) != 0'),

   # int64_t sum = a + b;
   #
   # if (a < 0 && b < 0 && a < sum)
   #    sum = INT64_MIN;
   # } else if (a >= 0 && b >= 0 && sum < a)
   #    sum = INT64_MAX;
   # }
   #
   # A couple optimizations are applied.
   #
   # 1. a < sum => sum >= 0.  This replacement works because it is known that
   #    a < 0 and b < 0, so sum should also be < 0 unless there was
   #    underflow.
   #
   # 2. sum < a => sum < 0.  This replacement works because it is known that
   #    a >= 0 and b >= 0, so sum should also be >= 0 unless there was
   #    overflow.
   #
   # 3. Invert the second if-condition and swap the order of parameters for
   #    the bcsel. !(a >= 0 && b >= 0 && sum < 0) becomes !(a >= 0) || !(b >=
   #    0) || !(sum < 0), and that becomes (a < 0) || (b < 0) || (sum >= 0)
   #
   # On Intel Gen11, this saves ~11 instructions.
   (('iadd_sat@64', a, b), ('bcsel',
                            ('iand', ('iand', ('ilt', a, 0), ('ilt', b, 0)), ('ige', ('iadd', a, b), 0)),
                            0x8000000000000000,
                            ('bcsel',
                             ('ior', ('ior', ('ilt', a, 0), ('ilt', b, 0)), ('ige', ('iadd', a, b), 0)),
                             ('iadd', a, b),
                             0x7fffffffffffffff)),
    '(options->lower_int64_options & nir_lower_iadd64) != 0'),

   # int64_t sum = a - b;
   #
   # if (a < 0 && b >= 0 && a < sum)
   #    sum = INT64_MIN;
   # } else if (a >= 0 && b < 0 && a >= sum)
   #    sum = INT64_MAX;
   # }
   #
   # Optimizations similar to the iadd_sat case are applied here.
   (('isub_sat@64', a, b), ('bcsel',
                            ('iand', ('iand', ('ilt', a, 0), ('ige', b, 0)), ('ige', ('isub', a, b), 0)),
                            0x8000000000000000,
                            ('bcsel',
                             ('ior', ('ior', ('ilt', a, 0), ('ige', b, 0)), ('ige', ('isub', a, b), 0)),
                             ('isub', a, b),
                             0x7fffffffffffffff)),
    '(options->lower_int64_options & nir_lower_iadd64) != 0'),

   # These are done here instead of in the backend because the int64 lowering
   # pass will make a mess of the patterns.  The first patterns are
   # conditioned on nir_lower_minmax64 because it was not clear that it was
   # always an improvement on platforms that have real int64 support.  No
   # shaders in shader-db hit this, so it was hard to say one way or the
   # other.
   (('ilt', ('imax(is_used_once)', 'a@64', 'b@64'), 0), ('ilt', ('imax', ('unpack_64_2x32_split_y', a), ('unpack_64_2x32_split_y', b)), 0), '(options->lower_int64_options & nir_lower_minmax64) != 0'),
   (('ilt', ('imin(is_used_once)', 'a@64', 'b@64'), 0), ('ilt', ('imin', ('unpack_64_2x32_split_y', a), ('unpack_64_2x32_split_y', b)), 0), '(options->lower_int64_options & nir_lower_minmax64) != 0'),
   (('ige', ('imax(is_used_once)', 'a@64', 'b@64'), 0), ('ige', ('imax', ('unpack_64_2x32_split_y', a), ('unpack_64_2x32_split_y', b)), 0), '(options->lower_int64_options & nir_lower_minmax64) != 0'),
   (('ige', ('imin(is_used_once)', 'a@64', 'b@64'), 0), ('ige', ('imin', ('unpack_64_2x32_split_y', a), ('unpack_64_2x32_split_y', b)), 0), '(options->lower_int64_options & nir_lower_minmax64) != 0'),
   (('ilt', 'a@64', 0), ('ilt', ('unpack_64_2x32_split_y', a), 0), '(options->lower_int64_options & nir_lower_icmp64) != 0'),
   (('ige', 'a@64', 0), ('ige', ('unpack_64_2x32_split_y', a), 0), '(options->lower_int64_options & nir_lower_icmp64) != 0'),

   (('ine', 'a@64', 0), ('ine', ('ior', ('unpack_64_2x32_split_x', a), ('unpack_64_2x32_split_y', a)), 0), '(options->lower_int64_options & nir_lower_icmp64) != 0'),
   (('ieq', 'a@64', 0), ('ieq', ('ior', ('unpack_64_2x32_split_x', a), ('unpack_64_2x32_split_y', a)), 0), '(options->lower_int64_options & nir_lower_icmp64) != 0'),
   # 0u < uint(a) <=> uint(a) != 0u
   (('ult', 0, 'a@64'), ('ine', ('ior', ('unpack_64_2x32_split_x', a), ('unpack_64_2x32_split_y', a)), 0), '(options->lower_int64_options & nir_lower_icmp64) != 0'),

   # Alternative lowering that doesn't rely on bfi.
   (('bitfield_insert', 'base', 'insert', 'offset', 'bits'),
    ('bcsel', ('ult', 31, 'bits'),
     'insert',
    (('ior',
     ('iand', 'base', ('inot', ('ishl', ('isub', ('ishl', 1, 'bits'), 1), 'offset'))),
     ('iand', ('ishl', 'insert', 'offset'), ('ishl', ('isub', ('ishl', 1, 'bits'), 1), 'offset'))))),
    'options->lower_bitfield_insert_to_shifts'),

   # Alternative lowering that uses bitfield_select.
   (('bitfield_insert', 'base', 'insert', 'offset', 'bits'),
    ('bcsel', ('ult', 31, 'bits'), 'insert',
              ('bitfield_select', ('bfm', 'bits', 'offset'), ('ishl', 'insert', 'offset'), 'base')),
    'options->lower_bitfield_insert_to_bitfield_select'),

   (('ibitfield_extract', 'value', 'offset', 'bits'),
    ('bcsel', ('ult', 31, 'bits'), 'value',
              ('ibfe', 'value', 'offset', 'bits')),
    'options->lower_bitfield_extract'),

   (('ubitfield_extract', 'value', 'offset', 'bits'),
    ('bcsel', ('ult', 31, 'bits'), 'value',
              ('ubfe', 'value', 'offset', 'bits')),
    'options->lower_bitfield_extract'),

   # Note that these opcodes are defined to only use the five least significant bits of 'offset' and 'bits'
   (('ubfe', 'value', 'offset', ('iand', 31, 'bits')), ('ubfe', 'value', 'offset', 'bits')),
   (('ubfe', 'value', ('iand', 31, 'offset'), 'bits'), ('ubfe', 'value', 'offset', 'bits')),
   (('ibfe', 'value', 'offset', ('iand', 31, 'bits')), ('ibfe', 'value', 'offset', 'bits')),
   (('ibfe', 'value', ('iand', 31, 'offset'), 'bits'), ('ibfe', 'value', 'offset', 'bits')),
   (('bfm', 'bits', ('iand', 31, 'offset')), ('bfm', 'bits', 'offset')),
   (('bfm', ('iand', 31, 'bits'), 'offset'), ('bfm', 'bits', 'offset')),

   (('ibitfield_extract', 'value', 'offset', 'bits'),
    ('bcsel', ('ieq', 0, 'bits'),
     0,
     ('ishr',
       ('ishl', 'value', ('isub', ('isub', 32, 'bits'), 'offset')),
       ('isub', 32, 'bits'))),
    'options->lower_bitfield_extract_to_shifts'),

   (('ubitfield_extract', 'value', 'offset', 'bits'),
    ('iand',
     ('ushr', 'value', 'offset'),
     ('bcsel', ('ieq', 'bits', 32),
      0xffffffff,
      ('isub', ('ishl', 1, 'bits'), 1))),
    'options->lower_bitfield_extract_to_shifts'),

   (('ifind_msb', 'value'),
    ('ufind_msb', ('bcsel', ('ilt', 'value', 0), ('inot', 'value'), 'value')),
    'options->lower_ifind_msb'),

   (('find_lsb', 'value'),
    ('ufind_msb', ('iand', 'value', ('ineg', 'value'))),
    'options->lower_find_lsb'),

   (('extract_i8', a, 'b@32'),
    ('ishr', ('ishl', a, ('imul', ('isub', 3, b), 8)), 24),
    'options->lower_extract_byte'),

   (('extract_u8', a, 'b@32'),
    ('iand', ('ushr', a, ('imul', b, 8)), 0xff),
    'options->lower_extract_byte'),

   (('extract_i16', a, 'b@32'),
    ('ishr', ('ishl', a, ('imul', ('isub', 1, b), 16)), 16),
    'options->lower_extract_word'),

   (('extract_u16', a, 'b@32'),
    ('iand', ('ushr', a, ('imul', b, 16)), 0xffff),
    'options->lower_extract_word'),

    (('pack_unorm_2x16', 'v'),
     ('pack_uvec2_to_uint',
        ('f2u32', ('fround_even', ('fmul', ('fsat', 'v'), 65535.0)))),
     'options->lower_pack_unorm_2x16'),

    (('pack_unorm_4x8', 'v'),
     ('pack_uvec4_to_uint',
        ('f2u32', ('fround_even', ('fmul', ('fsat', 'v'), 255.0)))),
     'options->lower_pack_unorm_4x8'),

    (('pack_snorm_2x16', 'v'),
     ('pack_uvec2_to_uint',
        ('f2i32', ('fround_even', ('fmul', ('fmin', 1.0, ('fmax', -1.0, 'v')), 32767.0)))),
     'options->lower_pack_snorm_2x16'),

    (('pack_snorm_4x8', 'v'),
     ('pack_uvec4_to_uint',
        ('f2i32', ('fround_even', ('fmul', ('fmin', 1.0, ('fmax', -1.0, 'v')), 127.0)))),
     'options->lower_pack_snorm_4x8'),

    (('unpack_unorm_2x16', 'v'),
     ('fdiv', ('u2f32', ('vec2', ('extract_u16', 'v', 0),
                                  ('extract_u16', 'v', 1))),
              65535.0),
     'options->lower_unpack_unorm_2x16'),

    (('unpack_unorm_4x8', 'v'),
     ('fdiv', ('u2f32', ('vec4', ('extract_u8', 'v', 0),
                                  ('extract_u8', 'v', 1),
                                  ('extract_u8', 'v', 2),
                                  ('extract_u8', 'v', 3))),
              255.0),
     'options->lower_unpack_unorm_4x8'),

    (('unpack_snorm_2x16', 'v'),
     ('fmin', 1.0, ('fmax', -1.0, ('fdiv', ('i2f', ('vec2', ('extract_i16', 'v', 0),
                                                            ('extract_i16', 'v', 1))),
                                           32767.0))),
     'options->lower_unpack_snorm_2x16'),

    (('unpack_snorm_4x8', 'v'),
     ('fmin', 1.0, ('fmax', -1.0, ('fdiv', ('i2f', ('vec4', ('extract_i8', 'v', 0),
                                                            ('extract_i8', 'v', 1),
                                                            ('extract_i8', 'v', 2),
                                                            ('extract_i8', 'v', 3))),
                                           127.0))),
     'options->lower_unpack_snorm_4x8'),

   (('pack_half_2x16_split', 'a@32', 'b@32'),
    ('ior', ('ishl', ('u2u32', ('f2f16', b)), 16), ('u2u32', ('f2f16', a))),
    'options->lower_pack_half_2x16_split'),

   (('unpack_half_2x16_split_x', 'a@32'),
    ('f2f32', ('u2u16', a)),
    'options->lower_unpack_half_2x16_split'),

   (('unpack_half_2x16_split_y', 'a@32'),
    ('f2f32', ('u2u16', ('ushr', a, 16))),
    'options->lower_unpack_half_2x16_split'),

   (('isign', a), ('imin', ('imax', a, -1), 1), 'options->lower_isign'),
   (('fsign', a), ('fsub', ('b2f', ('flt', 0.0, a)), ('b2f', ('flt', a, 0.0))), 'options->lower_fsign'),

   # Address/offset calculations:
   # Drivers supporting imul24 should use the nir_lower_amul() pass, this
   # rule converts everyone else to imul:
   (('amul', a, b), ('imul', a, b), '!options->has_imul24'),

   (('imad24_ir3', a, b, 0), ('imul24', a, b)),
   (('imad24_ir3', a, 0, c), (c)),
   (('imad24_ir3', a, 1, c), ('iadd', a, c)),

   # if first two srcs are const, crack apart the imad so constant folding
   # can clean up the imul:
   # TODO ffma should probably get a similar rule:
   (('imad24_ir3', '#a', '#b', c), ('iadd', ('imul', a, b), c)),

   # These will turn 24b address/offset calc back into 32b shifts, but
   # it should be safe to get back some of the bits of precision that we
   # already decided were no necessary:
   (('imul24', a, '#b@32(is_pos_power_of_two)'), ('ishl', a, ('find_lsb', b)), '!options->lower_bitops'),
   (('imul24', a, '#b@32(is_neg_power_of_two)'), ('ineg', ('ishl', a, ('find_lsb', ('iabs', b)))), '!options->lower_bitops'),
   (('imul24', a, 0), (0)),
])

# bit_size dependent lowerings
for bit_size in [8, 16, 32, 64]:
   # convenience constants
   intmax = (1 << (bit_size - 1)) - 1
   intmin = 1 << (bit_size - 1)

   optimizations += [
      (('iadd_sat@' + str(bit_size), a, b),
       ('bcsel', ('ige', b, 1), ('bcsel', ('ilt', ('iadd', a, b), a), intmax, ('iadd', a, b)),
                                ('bcsel', ('ilt', a, ('iadd', a, b)), intmin, ('iadd', a, b))), 'options->lower_add_sat'),
      (('isub_sat@' + str(bit_size), a, b),
       ('bcsel', ('ilt', b, 0), ('bcsel', ('ilt', ('isub', a, b), a), intmax, ('isub', a, b)),
                                ('bcsel', ('ilt', a, ('isub', a, b)), intmin, ('isub', a, b))), 'options->lower_add_sat'),
   ]

invert = OrderedDict([('feq', 'fne'), ('fne', 'feq')])

for left, right in itertools.combinations_with_replacement(invert.keys(), 2):
   optimizations.append((('inot', ('ior(is_used_once)', (left, a, b), (right, c, d))),
                         ('iand', (invert[left], a, b), (invert[right], c, d))))
   optimizations.append((('inot', ('iand(is_used_once)', (left, a, b), (right, c, d))),
                         ('ior', (invert[left], a, b), (invert[right], c, d))))

# Optimize x2bN(b2x(x)) -> x
for size in type_sizes('bool'):
    aN = 'a@' + str(size)
    f2bN = 'f2b' + str(size)
    i2bN = 'i2b' + str(size)
    optimizations.append(((f2bN, ('b2f', aN)), a))
    optimizations.append(((i2bN, ('b2i', aN)), a))

# Optimize x2yN(b2x(x)) -> b2y
for x, y in itertools.product(['f', 'u', 'i'], ['f', 'u', 'i']):
   if x != 'f' and y != 'f' and x != y:
      continue

   b2x = 'b2f' if x == 'f' else 'b2i'
   b2y = 'b2f' if y == 'f' else 'b2i'
   x2yN = '{}2{}'.format(x, y)
   optimizations.append(((x2yN, (b2x, a)), (b2y, a)))

# Optimize away x2xN(a@N)
for t in ['int', 'uint', 'float', 'bool']:
   for N in type_sizes(t):
      x2xN = '{0}2{0}{1}'.format(t[0], N)
      aN = 'a@{0}'.format(N)
      optimizations.append(((x2xN, aN), a))

# Optimize x2xN(y2yM(a@P)) -> y2yN(a) for integers
# In particular, we can optimize away everything except upcast of downcast and
# upcasts where the type differs from the other cast
for N, M in itertools.product(type_sizes('uint'), type_sizes('uint')):
   if N < M:
      # The outer cast is a down-cast.  It doesn't matter what the size of the
      # argument of the inner cast is because we'll never been in the upcast
      # of downcast case.  Regardless of types, we'll always end up with y2yN
      # in the end.
      for x, y in itertools.product(['i', 'u'], ['i', 'u']):
         x2xN = '{0}2{0}{1}'.format(x, N)
         y2yM = '{0}2{0}{1}'.format(y, M)
         y2yN = '{0}2{0}{1}'.format(y, N)
         optimizations.append(((x2xN, (y2yM, a)), (y2yN, a)))
   elif N > M:
      # If the outer cast is an up-cast, we have to be more careful about the
      # size of the argument of the inner cast and with types.  In this case,
      # the type is always the type of type up-cast which is given by the
      # outer cast.
      for P in type_sizes('uint'):
         # We can't optimize away up-cast of down-cast.
         if M < P:
            continue

         # Because we're doing down-cast of down-cast, the types always have
         # to match between the two casts
         for x in ['i', 'u']:
            x2xN = '{0}2{0}{1}'.format(x, N)
            x2xM = '{0}2{0}{1}'.format(x, M)
            aP = 'a@{0}'.format(P)
            optimizations.append(((x2xN, (x2xM, aP)), (x2xN, a)))
   else:
      # The N == M case is handled by other optimizations
      pass

# Downcast operations should be able to see through pack
for t in ['i', 'u']:
    for N in [8, 16, 32]:
        x2xN = '{0}2{0}{1}'.format(t, N)
        optimizations += [
            ((x2xN, ('pack_64_2x32_split', a, b)), (x2xN, a)),
            ((x2xN, ('pack_64_2x32_split', a, b)), (x2xN, a)),
        ]

# Optimize comparisons with up-casts
for t in ['int', 'uint', 'float']:
    for N, M in itertools.product(type_sizes(t), repeat=2):
        if N == 1 or N >= M:
            continue

        x2xM = '{0}2{0}{1}'.format(t[0], M)
        x2xN = '{0}2{0}{1}'.format(t[0], N)
        aN = 'a@' + str(N)
        bN = 'b@' + str(N)
        xeq = 'feq' if t == 'float' else 'ieq'
        xne = 'fne' if t == 'float' else 'ine'
        xge = '{0}ge'.format(t[0])
        xlt = '{0}lt'.format(t[0])

        # Up-casts are lossless so for correctly signed comparisons of
        # up-casted values we can do the comparison at the largest of the two
        # original sizes and drop one or both of the casts.  (We have
        # optimizations to drop the no-op casts which this may generate.)
        for P in type_sizes(t):
            if P == 1 or P > N:
                continue

            bP = 'b@' + str(P)
            optimizations += [
                ((xeq, (x2xM, aN), (x2xM, bP)), (xeq, a, (x2xN, b))),
                ((xne, (x2xM, aN), (x2xM, bP)), (xne, a, (x2xN, b))),
                ((xge, (x2xM, aN), (x2xM, bP)), (xge, a, (x2xN, b))),
                ((xlt, (x2xM, aN), (x2xM, bP)), (xlt, a, (x2xN, b))),
                ((xge, (x2xM, bP), (x2xM, aN)), (xge, (x2xN, b), a)),
                ((xlt, (x2xM, bP), (x2xM, aN)), (xlt, (x2xN, b), a)),
            ]

        # The next bit doesn't work on floats because the range checks would
        # get way too complicated.
        if t in ['int', 'uint']:
            if t == 'int':
                xN_min = -(1 << (N - 1))
                xN_max = (1 << (N - 1)) - 1
            elif t == 'uint':
                xN_min = 0
                xN_max = (1 << N) - 1
            else:
                assert False

            # If we're up-casting and comparing to a constant, we can unfold
            # the comparison into a comparison with the shrunk down constant
            # and a check that the constant fits in the smaller bit size.
            optimizations += [
                ((xeq, (x2xM, aN), '#b'),
                 ('iand', (xeq, a, (x2xN, b)), (xeq, (x2xM, (x2xN, b)), b))),
                ((xne, (x2xM, aN), '#b'),
                 ('ior', (xne, a, (x2xN, b)), (xne, (x2xM, (x2xN, b)), b))),
                ((xlt, (x2xM, aN), '#b'),
                 ('iand', (xlt, xN_min, b),
                          ('ior', (xlt, xN_max, b), (xlt, a, (x2xN, b))))),
                ((xlt, '#a', (x2xM, bN)),
                 ('iand', (xlt, a, xN_max),
                          ('ior', (xlt, a, xN_min), (xlt, (x2xN, a), b)))),
                ((xge, (x2xM, aN), '#b'),
                 ('iand', (xge, xN_max, b),
                          ('ior', (xge, xN_min, b), (xge, a, (x2xN, b))))),
                ((xge, '#a', (x2xM, bN)),
                 ('iand', (xge, a, xN_min),
                          ('ior', (xge, a, xN_max), (xge, (x2xN, a), b)))),
            ]

def fexp2i(exp, bits):
   # Generate an expression which constructs value 2.0^exp or 0.0.
   #
   # We assume that exp is already in a valid range:
   #
   #   * [-15, 15] for 16-bit float
   #   * [-127, 127] for 32-bit float
   #   * [-1023, 1023] for 16-bit float
   #
   # If exp is the lowest value in the valid range, a value of 0.0 is
   # constructed.  Otherwise, the value 2.0^exp is constructed.
   if bits == 16:
      return ('i2i16', ('ishl', ('iadd', exp, 15), 10))
   elif bits == 32:
      return ('ishl', ('iadd', exp, 127), 23)
   elif bits == 64:
      return ('pack_64_2x32_split', 0, ('ishl', ('iadd', exp, 1023), 20))
   else:
      assert False

def ldexp(f, exp, bits):
   # The maximum possible range for a normal exponent is [-126, 127] and,
   # throwing in denormals, you get a maximum range of [-149, 127].  This
   # means that we can potentially have a swing of +-276.  If you start with
   # FLT_MAX, you actually have to do ldexp(FLT_MAX, -278) to get it to flush
   # all the way to zero.  The GLSL spec only requires that we handle a subset
   # of this range.  From version 4.60 of the spec:
   #
   #    "If exp is greater than +128 (single-precision) or +1024
   #    (double-precision), the value returned is undefined. If exp is less
   #    than -126 (single-precision) or -1022 (double-precision), the value
   #    returned may be flushed to zero. Additionally, splitting the value
   #    into a significand and exponent using frexp() and then reconstructing
   #    a floating-point value using ldexp() should yield the original input
   #    for zero and all finite non-denormalized values."
   #
   # The SPIR-V spec has similar language.
   #
   # In order to handle the maximum value +128 using the fexp2i() helper
   # above, we have to split the exponent in half and do two multiply
   # operations.
   #
   # First, we clamp exp to a reasonable range.  Specifically, we clamp to
   # twice the full range that is valid for the fexp2i() function above.  If
   # exp/2 is the bottom value of that range, the fexp2i() expression will
   # yield 0.0f which, when multiplied by f, will flush it to zero which is
   # allowed by the GLSL and SPIR-V specs for low exponent values.  If the
   # value is clamped from above, then it must have been above the supported
   # range of the GLSL built-in and therefore any return value is acceptable.
   if bits == 16:
      exp = ('imin', ('imax', exp, -30), 30)
   elif bits == 32:
      exp = ('imin', ('imax', exp, -254), 254)
   elif bits == 64:
      exp = ('imin', ('imax', exp, -2046), 2046)
   else:
      assert False

   # Now we compute two powers of 2, one for exp/2 and one for exp-exp/2.
   # (We use ishr which isn't the same for -1, but the -1 case still works
   # since we use exp-exp/2 as the second exponent.)  While the spec
   # technically defines ldexp as f * 2.0^exp, simply multiplying once doesn't
   # work with denormals and doesn't allow for the full swing in exponents
   # that you can get with normalized values.  Instead, we create two powers
   # of two and multiply by them each in turn.  That way the effective range
   # of our exponent is doubled.
   pow2_1 = fexp2i(('ishr', exp, 1), bits)
   pow2_2 = fexp2i(('isub', exp, ('ishr', exp, 1)), bits)
   return ('fmul', ('fmul', f, pow2_1), pow2_2)

optimizations += [
   (('ldexp@16', 'x', 'exp'), ldexp('x', 'exp', 16), 'options->lower_ldexp'),
   (('ldexp@32', 'x', 'exp'), ldexp('x', 'exp', 32), 'options->lower_ldexp'),
   (('ldexp@64', 'x', 'exp'), ldexp('x', 'exp', 64), 'options->lower_ldexp'),
]

# Unreal Engine 4 demo applications open-codes bitfieldReverse()
def bitfield_reverse(u):
    step1 = ('ior', ('ishl', u, 16), ('ushr', u, 16))
    step2 = ('ior', ('ishl', ('iand', step1, 0x00ff00ff), 8), ('ushr', ('iand', step1, 0xff00ff00), 8))
    step3 = ('ior', ('ishl', ('iand', step2, 0x0f0f0f0f), 4), ('ushr', ('iand', step2, 0xf0f0f0f0), 4))
    step4 = ('ior', ('ishl', ('iand', step3, 0x33333333), 2), ('ushr', ('iand', step3, 0xcccccccc), 2))
    step5 = ('ior(many-comm-expr)', ('ishl', ('iand', step4, 0x55555555), 1), ('ushr', ('iand', step4, 0xaaaaaaaa), 1))

    return step5

optimizations += [(bitfield_reverse('x@32'), ('bitfield_reverse', 'x'), '!options->lower_bitfield_reverse')]

# For any float comparison operation, "cmp", if you have "a == a && a cmp b"
# then the "a == a" is redundant because it's equivalent to "a is not NaN"
# and, if a is a NaN then the second comparison will fail anyway.
for op in ['flt', 'fge', 'feq']:
   optimizations += [
      (('iand', ('feq', a, a), (op, a, b)), ('!' + op, a, b)),
      (('iand', ('feq', a, a), (op, b, a)), ('!' + op, b, a)),
   ]

# Add optimizations to handle the case where the result of a ternary is
# compared to a constant.  This way we can take things like
#
# (a ? 0 : 1) > 0
#
# and turn it into
#
# a ? (0 > 0) : (1 > 0)
#
# which constant folding will eat for lunch.  The resulting ternary will
# further get cleaned up by the boolean reductions above and we will be
# left with just the original variable "a".
for op in ['flt', 'fge', 'feq', 'fne',
           'ilt', 'ige', 'ieq', 'ine', 'ult', 'uge']:
   optimizations += [
      ((op, ('bcsel', 'a', '#b', '#c'), '#d'),
       ('bcsel', 'a', (op, 'b', 'd'), (op, 'c', 'd'))),
      ((op, '#d', ('bcsel', a, '#b', '#c')),
       ('bcsel', 'a', (op, 'd', 'b'), (op, 'd', 'c'))),
   ]


# For example, this converts things like
#
#    1 + mix(0, a - 1, condition)
#
# into
#
#    mix(1, (a-1)+1, condition)
#
# Other optimizations will rearrange the constants.
for op in ['fadd', 'fmul', 'iadd', 'imul']:
   optimizations += [
      ((op, ('bcsel(is_used_once)', a, '#b', c), '#d'), ('bcsel', a, (op, b, d), (op, c, d)))
   ]

# For derivatives in compute shaders, GLSL_NV_compute_shader_derivatives
# states:
#
#     If neither layout qualifier is specified, derivatives in compute shaders
#     return zero, which is consistent with the handling of built-in texture
#     functions like texture() in GLSL 4.50 compute shaders.
for op in ['fddx', 'fddx_fine', 'fddx_coarse',
           'fddy', 'fddy_fine', 'fddy_coarse']:
   optimizations += [
      ((op, 'a'), 0.0, 'info->stage == MESA_SHADER_COMPUTE && info->cs.derivative_group == DERIVATIVE_GROUP_NONE')
]

# Some optimizations for ir3-specific instructions.
optimizations += [
   # 'al * bl': If either 'al' or 'bl' is zero, return zero.
   (('umul_low', '#a(is_lower_half_zero)', 'b'), (0)),
   # '(ah * bl) << 16 + c': If either 'ah' or 'bl' is zero, return 'c'.
   (('imadsh_mix16', '#a@32(is_lower_half_zero)', 'b@32', 'c@32'), ('c')),
   (('imadsh_mix16', 'a@32', '#b@32(is_upper_half_zero)', 'c@32'), ('c')),
]

# These kinds of sequences can occur after nir_opt_peephole_select.
#
# NOTE: fadd is not handled here because that gets in the way of ffma
# generation in the i965 driver.  Instead, fadd and ffma are handled in
# late_optimizations.

for op in ['flrp']:
    optimizations += [
        (('bcsel', a, (op + '(is_used_once)', b, c, d), (op, b, c, e)), (op, b, c, ('bcsel', a, d, e))),
        (('bcsel', a, (op, b, c, d), (op + '(is_used_once)', b, c, e)), (op, b, c, ('bcsel', a, d, e))),
        (('bcsel', a, (op + '(is_used_once)', b, c, d), (op, b, e, d)), (op, b, ('bcsel', a, c, e), d)),
        (('bcsel', a, (op, b, c, d), (op + '(is_used_once)', b, e, d)), (op, b, ('bcsel', a, c, e), d)),
        (('bcsel', a, (op + '(is_used_once)', b, c, d), (op, e, c, d)), (op, ('bcsel', a, b, e), c, d)),
        (('bcsel', a, (op, b, c, d), (op + '(is_used_once)', e, c, d)), (op, ('bcsel', a, b, e), c, d)),
    ]

for op in ['fmul', 'iadd', 'imul', 'iand', 'ior', 'ixor', 'fmin', 'fmax', 'imin', 'imax', 'umin', 'umax']:
    optimizations += [
        (('bcsel', a, (op + '(is_used_once)', b, c), (op, b, 'd(is_not_const)')), (op, b, ('bcsel', a, c, d))),
        (('bcsel', a, (op + '(is_used_once)', b, 'c(is_not_const)'), (op, b, d)), (op, b, ('bcsel', a, c, d))),
        (('bcsel', a, (op, b, 'c(is_not_const)'), (op + '(is_used_once)', b, d)), (op, b, ('bcsel', a, c, d))),
        (('bcsel', a, (op, b, c), (op + '(is_used_once)', b, 'd(is_not_const)')), (op, b, ('bcsel', a, c, d))),
    ]

for op in ['fpow']:
    optimizations += [
        (('bcsel', a, (op + '(is_used_once)', b, c), (op, b, d)), (op, b, ('bcsel', a, c, d))),
        (('bcsel', a, (op, b, c), (op + '(is_used_once)', b, d)), (op, b, ('bcsel', a, c, d))),
        (('bcsel', a, (op + '(is_used_once)', b, c), (op, d, c)), (op, ('bcsel', a, b, d), c)),
        (('bcsel', a, (op, b, c), (op + '(is_used_once)', d, c)), (op, ('bcsel', a, b, d), c)),
    ]

for op in ['frcp', 'frsq', 'fsqrt', 'fexp2', 'flog2', 'fsign', 'fsin', 'fcos']:
    optimizations += [
        (('bcsel', a, (op + '(is_used_once)', b), (op, c)), (op, ('bcsel', a, b, c))),
        (('bcsel', a, (op, b), (op + '(is_used_once)', c)), (op, ('bcsel', a, b, c))),
    ]

# This section contains "late" optimizations that should be run before
# creating ffmas and calling regular optimizations for the final time.
# Optimizations should go here if they help code generation and conflict
# with the regular optimizations.
before_ffma_optimizations = [
   # Propagate constants down multiplication chains
   (('~fmul(is_used_once)', ('fmul(is_used_once)', 'a(is_not_const)', '#b'), 'c(is_not_const)'), ('fmul', ('fmul', a, c), b)),
   (('imul(is_used_once)', ('imul(is_used_once)', 'a(is_not_const)', '#b'), 'c(is_not_const)'), ('imul', ('imul', a, c), b)),
   (('~fadd(is_used_once)', ('fadd(is_used_once)', 'a(is_not_const)', '#b'), 'c(is_not_const)'), ('fadd', ('fadd', a, c), b)),
   (('iadd(is_used_once)', ('iadd(is_used_once)', 'a(is_not_const)', '#b'), 'c(is_not_const)'), ('iadd', ('iadd', a, c), b)),

   (('~fadd', ('fmul', a, b), ('fmul', a, c)), ('fmul', a, ('fadd', b, c))),
   (('iadd', ('imul', a, b), ('imul', a, c)), ('imul', a, ('iadd', b, c))),
   (('~fadd', ('fneg', a), a), 0.0),
   (('iadd', ('ineg', a), a), 0),
   (('iadd', ('ineg', a), ('iadd', a, b)), b),
   (('iadd', a, ('iadd', ('ineg', a), b)), b),
   (('~fadd', ('fneg', a), ('fadd', a, b)), b),
   (('~fadd', a, ('fadd', ('fneg', a), b)), b),

   (('~flrp@32', ('fadd(is_used_once)', a, -1.0), ('fadd(is_used_once)', a,  1.0), d), ('fadd', ('flrp', -1.0,  1.0, d), a)),
   (('~flrp@32', ('fadd(is_used_once)', a,  1.0), ('fadd(is_used_once)', a, -1.0), d), ('fadd', ('flrp',  1.0, -1.0, d), a)),
   (('~flrp@32', ('fadd(is_used_once)', a, '#b'), ('fadd(is_used_once)', a, '#c'), d), ('fadd', ('fmul', d, ('fadd', c, ('fneg', b))), ('fadd', a, b))),
]

# This section contains "late" optimizations that should be run after the
# regular optimizations have finished.  Optimizations should go here if
# they help code generation but do not necessarily produce code that is
# more easily optimizable.
late_optimizations = [
   # Most of these optimizations aren't quite safe when you get infinity or
   # Nan involved but the first one should be fine.
   (('flt',          ('fadd', a, b),  0.0), ('flt',          a, ('fneg', b))),
   (('flt', ('fneg', ('fadd', a, b)), 0.0), ('flt', ('fneg', a),         b)),
   (('~fge',          ('fadd', a, b),  0.0), ('fge',          a, ('fneg', b))),
   (('~fge', ('fneg', ('fadd', a, b)), 0.0), ('fge', ('fneg', a),         b)),
   (('~feq', ('fadd', a, b), 0.0), ('feq', a, ('fneg', b))),
   (('~fne', ('fadd', a, b), 0.0), ('fne', a, ('fneg', b))),

   # nir_lower_to_source_mods will collapse this, but its existence during the
   # optimization loop can prevent other optimizations.
   (('fneg', ('fneg', a)), a),

   # Subtractions get lowered during optimization, so we need to recombine them
   (('fadd', 'a', ('fneg', 'b')), ('fsub', 'a', 'b'), '!options->lower_sub'),
   (('iadd', 'a', ('ineg', 'b')), ('isub', 'a', 'b'), '!options->lower_sub'),
   (('fneg', a), ('fsub', 0.0, a), 'options->lower_negate'),
   (('ineg', a), ('isub', 0, a), 'options->lower_negate'),

   # These are duplicated from the main optimizations table.  The late
   # patterns that rearrange expressions like x - .5 < 0 to x < .5 can create
   # new patterns like these.  The patterns that compare with zero are removed
   # because they are unlikely to be created in by anything in
   # late_optimizations.
   (('flt', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('flt', a, b)),
   (('flt', '#b(is_gt_0_and_lt_1)', ('fsat(is_used_once)', a)), ('flt', b, a)),
   (('fge', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('fge', a, b)),
   (('fge', '#b(is_gt_0_and_lt_1)', ('fsat(is_used_once)', a)), ('fge', b, a)),
   (('feq', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('feq', a, b)),
   (('fne', ('fsat(is_used_once)', a), '#b(is_gt_0_and_lt_1)'), ('fne', a, b)),

   (('fge', ('fsat(is_used_once)', a), 1.0), ('fge', a, 1.0)),
   (('flt', ('fsat(is_used_once)', a), 1.0), ('flt', a, 1.0)),

   (('~fge', ('fmin(is_used_once)', ('fadd(is_used_once)', a, b), ('fadd', c, d)), 0.0), ('iand', ('fge', a, ('fneg', b)), ('fge', c, ('fneg', d)))),

   (('flt', ('fneg', a), ('fneg', b)), ('flt', b, a)),
   (('fge', ('fneg', a), ('fneg', b)), ('fge', b, a)),
   (('feq', ('fneg', a), ('fneg', b)), ('feq', b, a)),
   (('fne', ('fneg', a), ('fneg', b)), ('fne', b, a)),
   (('flt', ('fneg', a), -1.0), ('flt', 1.0, a)),
   (('flt', -1.0, ('fneg', a)), ('flt', a, 1.0)),
   (('fge', ('fneg', a), -1.0), ('fge', 1.0, a)),
   (('fge', -1.0, ('fneg', a)), ('fge', a, 1.0)),
   (('fne', ('fneg', a), -1.0), ('fne', 1.0, a)),
   (('feq', -1.0, ('fneg', a)), ('feq', a, 1.0)),

   (('ior', a, a), a),
   (('iand', a, a), a),

   (('iand', ('ine(is_used_once)', 'a@32', 0), ('ine', 'b@32', 0)), ('ine', ('umin', a, b), 0)),
   (('ior',  ('ieq(is_used_once)', 'a@32', 0), ('ieq', 'b@32', 0)), ('ieq', ('umin', a, b), 0)),

   (('~fadd', ('fneg(is_used_once)', ('fsat(is_used_once)', 'a(is_not_fmul)')), 1.0), ('fsat', ('fadd', 1.0, ('fneg', a)))),

   (('fdot2', a, b), ('fdot_replicated2', a, b), 'options->fdot_replicates'),
   (('fdot3', a, b), ('fdot_replicated3', a, b), 'options->fdot_replicates'),
   (('fdot4', a, b), ('fdot_replicated4', a, b), 'options->fdot_replicates'),
   (('fdph', a, b), ('fdph_replicated', a, b), 'options->fdot_replicates'),

   (('~flrp@32', ('fadd(is_used_once)', a, b), ('fadd(is_used_once)', a, c), d), ('fadd', ('flrp', b, c, d), a)),
   (('~flrp@64', ('fadd(is_used_once)', a, b), ('fadd(is_used_once)', a, c), d), ('fadd', ('flrp', b, c, d), a)),

   (('~fadd@32', 1.0, ('fmul(is_used_once)', c , ('fadd', b, -1.0 ))), ('fadd', ('fadd', 1.0, ('fneg', c)), ('fmul', b, c)), 'options->lower_flrp32'),
   (('~fadd@64', 1.0, ('fmul(is_used_once)', c , ('fadd', b, -1.0 ))), ('fadd', ('fadd', 1.0, ('fneg', c)), ('fmul', b, c)), 'options->lower_flrp64'),

   # A similar operation could apply to any ffma(#a, b, #(-a/2)), but this
   # particular operation is common for expanding values stored in a texture
   # from [0,1] to [-1,1].
   (('~ffma@32', a,  2.0, -1.0), ('flrp', -1.0,  1.0,          a ), '!options->lower_flrp32'),
   (('~ffma@32', a, -2.0, -1.0), ('flrp', -1.0,  1.0, ('fneg', a)), '!options->lower_flrp32'),
   (('~ffma@32', a, -2.0,  1.0), ('flrp',  1.0, -1.0,          a ), '!options->lower_flrp32'),
   (('~ffma@32', a,  2.0,  1.0), ('flrp',  1.0, -1.0, ('fneg', a)), '!options->lower_flrp32'),
   (('~fadd@32', ('fmul(is_used_once)',  2.0, a), -1.0), ('flrp', -1.0,  1.0,          a ), '!options->lower_flrp32'),
   (('~fadd@32', ('fmul(is_used_once)', -2.0, a), -1.0), ('flrp', -1.0,  1.0, ('fneg', a)), '!options->lower_flrp32'),
   (('~fadd@32', ('fmul(is_used_once)', -2.0, a),  1.0), ('flrp',  1.0, -1.0,          a ), '!options->lower_flrp32'),
   (('~fadd@32', ('fmul(is_used_once)',  2.0, a),  1.0), ('flrp',  1.0, -1.0, ('fneg', a)), '!options->lower_flrp32'),

    # flrp(a, b, a)
    # a*(1-a) + b*a
    # a + -a*a + a*b    (1)
    # a + a*(b - a)
    # Option 1: ffma(a, (b-a), a)
    #
    # Alternately, after (1):
    # a*(1+b) + -a*a
    # a*((1+b) + -a)
    #
    # Let b=1
    #
    # Option 2: ffma(a, 2, -(a*a))
    # Option 3: ffma(a, 2, (-a)*a)
    # Option 4: ffma(a, -a, (2*a)
    # Option 5: a * (2 - a)
    #
    # There are a lot of other possible combinations.
   (('~ffma@32', ('fadd', b, ('fneg', a)), a, a), ('flrp', a, b, a), '!options->lower_flrp32'),
   (('~ffma@32', a, 2.0, ('fneg', ('fmul', a, a))), ('flrp', a, 1.0, a), '!options->lower_flrp32'),
   (('~ffma@32', a, 2.0, ('fmul', ('fneg', a), a)), ('flrp', a, 1.0, a), '!options->lower_flrp32'),
   (('~ffma@32', a, ('fneg', a), ('fmul', 2.0, a)), ('flrp', a, 1.0, a), '!options->lower_flrp32'),
   (('~fmul@32', a, ('fadd', 2.0, ('fneg', a))),    ('flrp', a, 1.0, a), '!options->lower_flrp32'),

   # we do these late so that we don't get in the way of creating ffmas
   (('fmin', ('fadd(is_used_once)', '#c', a), ('fadd(is_used_once)', '#c', b)), ('fadd', c, ('fmin', a, b))),
   (('fmax', ('fadd(is_used_once)', '#c', a), ('fadd(is_used_once)', '#c', b)), ('fadd', c, ('fmax', a, b))),

   (('bcsel', a, 0, ('b2f32', ('inot', 'b@bool'))), ('b2f32', ('inot', ('ior', a, b)))),

   # Putting this in 'optimizations' interferes with the bcsel(a, op(b, c),
   # op(b, d)) => op(b, bcsel(a, c, d)) transformations.  I do not know why.
   (('bcsel', ('feq', ('fsqrt', 'a(is_not_negative)'), 0.0), intBitsToFloat(0x7f7fffff), ('frsq', a)),
    ('fmin', ('frsq', a), intBitsToFloat(0x7f7fffff))),

   # Things that look like DPH in the source shader may get expanded to
   # something that looks like dot(v1.xyz, v2.xyz) + v1.w by the time it gets
   # to NIR.  After FFMA is generated, this can look like:
   #
   #    fadd(ffma(v1.z, v2.z, ffma(v1.y, v2.y, fmul(v1.x, v2.x))), v1.w)
   #
   # Reassociate the last addition into the first multiplication.
   #
   # Some shaders do not use 'invariant' in vertex and (possibly) geometry
   # shader stages on some outputs that are intended to be invariant.  For
   # various reasons, this optimization may not be fully applied in all
   # shaders used for different rendering passes of the same geometry.  This
   # can result in Z-fighting artifacts (at best).  For now, disable this
   # optimization in these stages.  See bugzilla #111490.  In tessellation
   # stages applications seem to use 'precise' when necessary, so allow the
   # optimization in those stages.
   (('~fadd', ('ffma(is_used_once)', a, b, ('ffma', c, d, ('fmul', 'e(is_not_const_and_not_fsign)', 'f(is_not_const_and_not_fsign)'))), 'g(is_not_const)'),
    ('ffma', a, b, ('ffma', c, d, ('ffma', e, 'f', 'g'))), '(info->stage != MESA_SHADER_VERTEX && info->stage != MESA_SHADER_GEOMETRY) && !options->intel_vec4'),
   (('~fadd', ('ffma(is_used_once)', a, b, ('fmul', 'c(is_not_const_and_not_fsign)', 'd(is_not_const_and_not_fsign)') ), 'e(is_not_const)'),
    ('ffma', a, b, ('ffma', c, d, e)), '(info->stage != MESA_SHADER_VERTEX && info->stage != MESA_SHADER_GEOMETRY) && !options->intel_vec4'),

   # Convert f2fmp instructions to concrete f2f16 instructions. At this point
   # any conversions that could have been removed will have been removed in
   # nir_opt_algebraic so any remaining ones are required.
   (('f2fmp', a), ('f2f16', a)),
]

for op in ['fadd']:
    late_optimizations += [
        (('bcsel', a, (op + '(is_used_once)', b, c), (op, b, d)), (op, b, ('bcsel', a, c, d))),
        (('bcsel', a, (op, b, c), (op + '(is_used_once)', b, d)), (op, b, ('bcsel', a, c, d))),
    ]

for op in ['ffma']:
    late_optimizations += [
        (('bcsel', a, (op + '(is_used_once)', b, c, d), (op, b, c, e)), (op, b, c, ('bcsel', a, d, e))),
        (('bcsel', a, (op, b, c, d), (op + '(is_used_once)', b, c, e)), (op, b, c, ('bcsel', a, d, e))),

        (('bcsel', a, (op + '(is_used_once)', b, c, d), (op, b, e, d)), (op, b, ('bcsel', a, c, e), d)),
        (('bcsel', a, (op, b, c, d), (op + '(is_used_once)', b, e, d)), (op, b, ('bcsel', a, c, e), d)),
    ]

distribute_src_mods = [
   # Try to remove some spurious negations rather than pushing them down.
   (('fmul', ('fneg', a), ('fneg', b)), ('fmul', a, b)),
   (('ffma', ('fneg', a), ('fneg', b), c), ('ffma', a, b, c)),
   (('fdot_replicated2', ('fneg', a), ('fneg', b)), ('fdot_replicated2', a, b)),
   (('fdot_replicated3', ('fneg', a), ('fneg', b)), ('fdot_replicated3', a, b)),
   (('fdot_replicated4', ('fneg', a), ('fneg', b)), ('fdot_replicated4', a, b)),
   (('fneg', ('fneg', a)), a),

   (('fneg', ('ffma(is_used_once)', a, b, c)), ('ffma', ('fneg', a), b, ('fneg', c))),
   (('fneg', ('flrp(is_used_once)', a, b, c)), ('flrp', ('fneg', a), ('fneg', b), c)),
   (('fneg', ('fadd(is_used_once)', a, b)), ('fadd', ('fneg', a), ('fneg', b))),

   # Note that fmin <-> fmax.  I don't think there is a way to distribute
   # fabs() into fmin or fmax.
   (('fneg', ('fmin(is_used_once)', a, b)), ('fmax', ('fneg', a), ('fneg', b))),
   (('fneg', ('fmax(is_used_once)', a, b)), ('fmin', ('fneg', a), ('fneg', b))),

   # fdph works mostly like fdot, but to get the correct result, the negation
   # must be applied to the second source.
   (('fneg', ('fdph_replicated(is_used_once)', a, b)), ('fdph_replicated', a, ('fneg', b))),
   (('fabs', ('fdph_replicated(is_used_once)', a, b)), ('fdph_replicated', ('fabs', a), ('fabs', b))),

   (('fneg', ('fsign(is_used_once)', a)), ('fsign', ('fneg', a))),
   (('fabs', ('fsign(is_used_once)', a)), ('fsign', ('fabs', a))),
]

for op in ['fmul', 'fdot_replicated2', 'fdot_replicated3', 'fdot_replicated4']:
   distribute_src_mods.extend([
       (('fneg', (op + '(is_used_once)', a, b)), (op, ('fneg', a), b)),
       (('fabs', (op + '(is_used_once)', a, b)), (op, ('fabs', a), ('fabs', b))),
   ])

print(nir_algebraic.AlgebraicPass("nir_opt_algebraic", optimizations).render())
print(nir_algebraic.AlgebraicPass("nir_opt_algebraic_before_ffma",
                                  before_ffma_optimizations).render())
print(nir_algebraic.AlgebraicPass("nir_opt_algebraic_late",
                                  late_optimizations).render())
print(nir_algebraic.AlgebraicPass("nir_opt_algebraic_distribute_src_mods",
                                  distribute_src_mods).render())