ShiftDivMod. Implement faster divmod() for moduli with trailing 0 bits

Info:	This is the README file for ShiftDivMod.
Author:	Shlomi Fish <shlomif@cpan.org>
Copyright:	© 2020, Shlomi Fish.
Date:	2020-09-20
Version:	0.4.3

PURPOSE

This distribution implements faster divmod() (and .mod()) operations for moduli with a large number of trailing 0 bits (where the div/mod base is divisible by 2 ** n for an integer n).

It should yield the same result as the built-n divmod() function for positive numerators (its behaviour for negative ones is currently untested and undefined).

Also provided is a port to C and gmplib (= GNU multiple precision): https://github.com/shlomif/shift_divmod/tree/master/gmp-shift_divmod

INSTALLATION

pip3 install shift_divmod

USAGE

from shift_divmod import ShiftDivMod

base = 997
shift = 1200
modder = ShiftDivMod(base, shift)
# Alternative constructor which may require more
# work and eventualy calls the default constructor
modder = ShiftDivMod.from_int(base << shift)

x = 10 ** 500
# Same as divmod(x, (base << shift))
print( modder.divmod(x) )

NOTES

The code from which this distribution has been derived, was proposed as a proof-of-concept for a potential improvement for the built in cpython3 operations here: https://bugs.python.org/issue41487 . However, changing cpython3 in this manner was rejected.

libdivide ( https://github.com/ridiculousfish/libdivide ) provides a different, but also interesting, approach for optimizing division.

BENCHMARKS:

On my system, I got these results after running python3 code/examples/shift_divmod_example.py bench (reformated for clarity):

{'val': 5206685, 'time': 38.86349368095398, 'reached': 1000,
 'interrupted': False, 'mode': 'gen_shift_mod'}
{'val': 5206685, 'time': 39.018390417099, 'reached': 1000,
 'interrupted': False, 'mode': 'shiftmodpre'}
{'val': mpz(5206685), 'time': 167.4433994293213, 'reached': 1000,
 'interrupted': False, 'mode': 'gmpy'}
{'val': 3346424, 'time': 229.94409656524658, 'reached': 25,
 'interrupted': True, 'mode': 'builtinops'}

System:    Kernel: 5.8.8-200.fc32.x86_64 x86_64 bits: 64
    Desktop: KDE Plasma 5.18.5
           Distro: Fedora release 32 (Thirty Two)
CPU:       Info: Quad Core model: Intel Core i5-8259U
    bits: 64 type: MT MCP L2 cache: 6144 KiB
           Speed: 1600 MHz min/max: 400/3800 MHz Core speeds (MHz):
                1: 1600 2: 1600 3: 1601
           4: 1600 5: 1600 6: 1601 7: 1601 8: 1601
Graphics:  Device-1: Intel Iris Plus Graphics 655 driver: i915 v: kernel
           Display: server: Fedora Project
           X.org 1.20.8 driver: modesetting unloaded: fbdev,vesa
           resolution: 1920x1080~60Hz
           OpenGL: renderer: Mesa Intel Iris Plus
           Graphics 655 (CFL GT3) v: 4.6 Mesa 20.1.7

As can be noticed the shift_divmod based versions are over 4 times faster than GMP and much faster than the builtinops which only completed 25 out of 1,000 iterations before being interrupted. Note that for that use case, using GMP's modular exponentiation seems even faster.

With the C and gmplib version:

• mpz_mod runs the benchmark in about 160 seconds.
• shift_divmod runs the benchmark in about 35 seconds.
• pypy3 runs the pure-Python code in 25 seconds (!).

The Secret Sauce:

The code utilises the fact that bitwise operations are fast, and the magic happens in this code (with some comments added for clarity):

# Precalculating the object's field so that:
# self.shift : a non-negative integer signifying the bit shift
# self.base  : a non-negative integer which is shifted to
# form the modulu
# self.n = self.base << self.shift
# self.mask = ((1 << self.shift) - 1)
def divmod(self, inp):
    """calculate divmod(inp, self.n)"""
    if inp < self.n:
        return 0, inp
    div, mod = divmod((inp >> self.shift), self.base)
    return div, ((mod << self.shift) | (inp & self.mask))

(Or the equivalent but more bureaucratic C and gmplib code.)

COPYRIGHT

Copyright © 2020, Shlomi Fish. All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this list of conditions, and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions, and the following disclaimer in the documentation and/or other materials provided with the distribution.
3. Neither the name of the author of this software nor the names of contributors to this software may be used to endorse or promote products derived from this software without specific prior written consent.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.