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# Copyright (C) 2020 Arm Limited or its affiliates. All rights reserved.
#
# Copyright 2015 The Gemmlowp Authors. All Rights Reserved.
#
# SPDX-License-Identifier: Apache-2.0
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#
# Description:
# Contains various fixed point math functions based on the gemmlowp fixed
# point implementation.
import numpy as np
# Convert floating point to fixed point, default Q5.26
def from_float(x, integer_bits=5):
i32info = np.iinfo(np.int32)
fractional_bits = i32info.bits - integer_bits - 1
return min(max(round(x * (1 << fractional_bits)), i32info.min), i32info.max)
# Convert fixed point to floating point, default Q5.26
def to_float(x, integer_bits=5):
fractional_bits = np.iinfo(np.int32).bits - integer_bits - 1
return x / (1 << fractional_bits)
def saturating_rounding_mul32(a, b):
assert np.int32(a) == a
assert np.int32(b) == b
if a == b and a == np.iinfo(np.int32).min:
return np.int32(np.iinfo(np.int32).max)
divider = 1 << 31
ab = np.int64(a) * np.int64(b)
if ab >= 0:
nudge = 1 << 30
return (ab + nudge) // divider
else:
nudge = 1 - (1 << 30)
ab_plus_nudge = ab + nudge
result = ab_plus_nudge // divider
# Python uses floor, the reference uses truncation
# so we need to compensate for that.
if result * divider < ab_plus_nudge:
result += 1
return result
def saturating_rounding_mul16(a, b):
assert np.int16(a) == a
assert np.int16(b) == b
if a == b and a == np.iinfo(np.int16).min:
return np.int16(np.iinfo(np.int16).max)
divider = 1 << 15
ab = np.int32(a) * np.int32(b)
if ab >= 0:
nudge = 1 << 14
return (ab + nudge) // divider
else:
nudge = 1 - (1 << 14)
ab_plus_nudge = ab + nudge
result = ab_plus_nudge // divider
# Python uses floor, the reference uses truncation
# so we need to compensate for that.
if result * divider < ab_plus_nudge:
result += 1
return result
# Similar to saturating_rounding_mul16 except rounding to zero instead of to nearest
# Only supports 16bit
def saturating_mul16(a, b):
assert np.int16(a) == a
assert np.int16(b) == b
if a == b and a == np.iinfo(np.int16).min:
return np.int16(np.iinfo(np.int16).max)
ab = np.int32(a) * np.int32(b)
divider = 1 << 15
if ab >= 0:
return ab // divider
else:
result = ab // divider
# Python uses floor, the reference uses truncation
# so we need to compensate for that.
if result * divider < ab:
result += 1
return result
def shift_left32(a, offset):
assert offset >= 0
assert np.int32(a) == a
shifted = a * (1 << offset)
if shifted < np.iinfo(np.int32).min:
return np.int32(np.iinfo(np.int32).min)
elif shifted > np.iinfo(np.int32).max:
return np.int32(np.iinfo(np.int32).max)
else:
return np.int32(shifted)
def shift_left16(a, offset):
assert offset >= 0
assert np.int16(a) == a
shifted = a * (1 << offset)
if shifted < np.iinfo(np.int16).min:
return np.int16(np.iinfo(np.int16).min)
elif shifted > np.iinfo(np.int16).max:
return np.int16(np.iinfo(np.int16).max)
else:
return np.int16(shifted)
def downscale_multiplier_int32_to_int16(a):
assert np.int32(a) == a
rounding_offset = 1 << 15
if a >= np.iinfo(np.int32).max - rounding_offset:
return np.iinfo(np.int16).max
else:
return np.int16((a + rounding_offset) >> 16)
def rounding_divide_by_pot(x, exponent):
assert np.int32(x) == x
assert np.int32(exponent) == exponent
mask = (1 << exponent) - 1
remainder = x & mask
threshold = mask >> 1
if x < 0:
threshold += 1
result = x >> exponent
if remainder > threshold:
result += 1
return result
def saturating_rounding_multiply_by_pot(x, exponent):
assert np.int32(x) == x
assert np.int32(exponent) == exponent
threshold = (1 << (np.iinfo(np.int32).bits - 1 - exponent)) - 1
if x > threshold:
return np.iinfo(np.int32).max
elif x < -threshold:
return np.iinfo(np.int32).min
else:
return shift_left32(x, exponent)
def rescale(integer_bits_src, integer_bits_dst, x):
assert np.int32(integer_bits_src) == integer_bits_src
assert np.int32(integer_bits_dst) == integer_bits_dst
assert np.int32(x) == x
exponent = integer_bits_src - integer_bits_dst
if exponent < 0:
result = rounding_divide_by_pot(x, -exponent)
else:
result = saturating_rounding_multiply_by_pot(x, exponent)
return result
# Input Q0.31
def exp_on_interval_between_negative_one_quarter_and_0_excl(a):
assert np.int32(a) == a
assert -1 << (31 - 2) <= a < 0
offset = 28
constant_term = 1895147668
constant_1_over_3 = 715827883
x = a + (1 << offset)
x2 = saturating_rounding_mul32(x, x)
x3 = saturating_rounding_mul32(x2, x)
x4 = saturating_rounding_mul32(x2, x2)
x4_over_4 = rounding_divide_by_pot(x4, 2)
x4_over_24_plus_x3_over_6_plus_x2_over_2 = rounding_divide_by_pot(
saturating_rounding_mul32((x4_over_4 + x3), constant_1_over_3) + x2, 1
)
return np.int32(
constant_term + saturating_rounding_mul32(constant_term, x + x4_over_24_plus_x3_over_6_plus_x2_over_2)
)
# Input Q5.26
def exp_on_negative_values(a):
assert np.int32(a) == a
assert a <= 0
one_quarter = np.int32(16777216)
mask = np.int32(16777215)
a_mod_quarter_minus_one_quarter = np.int32((a & mask) - one_quarter)
result = exp_on_interval_between_negative_one_quarter_and_0_excl(rescale(5, 0, a_mod_quarter_minus_one_quarter))
remainder = np.int32(a_mod_quarter_minus_one_quarter - a)
def exp_barrel_shifter(exponent, multiplier, result):
fractional_bits = 26
integer_bits = 5
shift = fractional_bits + exponent if integer_bits > exponent else 0
if remainder & (1 << shift):
return saturating_rounding_mul32(result, multiplier)
else:
return result
result = exp_barrel_shifter(-2, 1672461947, result)
result = exp_barrel_shifter(-1, 1302514674, result)
result = exp_barrel_shifter(+0, 790015084, result)
result = exp_barrel_shifter(+1, 290630308, result)
result = exp_barrel_shifter(+2, 39332535, result)
result = exp_barrel_shifter(+3, 720401, result)
result = exp_barrel_shifter(+4, 242, result)
if a == 0:
return np.iinfo(np.int32).max
else:
return result
def multiply_by_quantized_multiplier(x, scale, shift):
# Multiplies x (int32) by (scale, shift) which have obtained by a call to scaling.quantize_scale,
# returns rounded result
shift = 31 - shift
left_shift = shift if shift > 0 else 0
right_shift = -shift if shift < 0 else 0
mul = saturating_rounding_mul32(x * (1 << left_shift), scale)
return rounding_divide_by_pot(mul, right_shift)
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