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/*-
* BSD LICENSE
*
* Copyright(c) 2010-2014 Intel Corporation. All rights reserved.
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* * 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.
* * Neither the name of Intel Corporation nor the names of its
* contributors may be used to endorse or promote products derived
* from this software without specific prior written permission.
*
* 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.
*/
#include <stdlib.h>
#include "rte_approx.h"
/*
* Based on paper "Approximating Rational Numbers by Fractions" by Michal
* Forisek forisek@dcs.fmph.uniba.sk
*
* Given a rational number alpha with 0 < alpha < 1 and a precision d, the goal
* is to find positive integers p, q such that alpha - d < p/q < alpha + d, and
* q is minimal.
*
* http://people.ksp.sk/~misof/publications/2007approx.pdf
*/
/* fraction comparison: compare (a/b) and (c/d) */
static inline uint32_t
less(uint32_t a, uint32_t b, uint32_t c, uint32_t d)
{
return a*d < b*c;
}
static inline uint32_t
less_or_equal(uint32_t a, uint32_t b, uint32_t c, uint32_t d)
{
return a*d <= b*c;
}
/* check whether a/b is a valid approximation */
static inline uint32_t
matches(uint32_t a, uint32_t b,
uint32_t alpha_num, uint32_t d_num, uint32_t denum)
{
if (less_or_equal(a, b, alpha_num - d_num, denum))
return 0;
if (less(a ,b, alpha_num + d_num, denum))
return 1;
return 0;
}
static inline void
find_exact_solution_left(uint32_t p_a, uint32_t q_a, uint32_t p_b, uint32_t q_b,
uint32_t alpha_num, uint32_t d_num, uint32_t denum, uint32_t *p, uint32_t *q)
{
uint32_t k_num = denum * p_b - (alpha_num + d_num) * q_b;
uint32_t k_denum = (alpha_num + d_num) * q_a - denum * p_a;
uint32_t k = (k_num / k_denum) + 1;
*p = p_b + k * p_a;
*q = q_b + k * q_a;
}
static inline void
find_exact_solution_right(uint32_t p_a, uint32_t q_a, uint32_t p_b, uint32_t q_b,
uint32_t alpha_num, uint32_t d_num, uint32_t denum, uint32_t *p, uint32_t *q)
{
uint32_t k_num = - denum * p_b + (alpha_num - d_num) * q_b;
uint32_t k_denum = - (alpha_num - d_num) * q_a + denum * p_a;
uint32_t k = (k_num / k_denum) + 1;
*p = p_b + k * p_a;
*q = q_b + k * q_a;
}
static int
find_best_rational_approximation(uint32_t alpha_num, uint32_t d_num, uint32_t denum, uint32_t *p, uint32_t *q)
{
uint32_t p_a, q_a, p_b, q_b;
/* check assumptions on the inputs */
if (!((0 < d_num) && (d_num < alpha_num) && (alpha_num < denum) && (d_num + alpha_num < denum))) {
return -1;
}
/* set initial bounds for the search */
p_a = 0;
q_a = 1;
p_b = 1;
q_b = 1;
while (1) {
uint32_t new_p_a, new_q_a, new_p_b, new_q_b;
uint32_t x_num, x_denum, x;
int aa, bb;
/* compute the number of steps to the left */
x_num = denum * p_b - alpha_num * q_b;
x_denum = - denum * p_a + alpha_num * q_a;
x = (x_num + x_denum - 1) / x_denum; /* x = ceil(x_num / x_denum) */
/* check whether we have a valid approximation */
aa = matches(p_b + x * p_a, q_b + x * q_a, alpha_num, d_num, denum);
bb = matches(p_b + (x-1) * p_a, q_b + (x - 1) * q_a, alpha_num, d_num, denum);
if (aa || bb) {
find_exact_solution_left(p_a, q_a, p_b, q_b, alpha_num, d_num, denum, p, q);
return 0;
}
/* update the interval */
new_p_a = p_b + (x - 1) * p_a ;
new_q_a = q_b + (x - 1) * q_a;
new_p_b = p_b + x * p_a ;
new_q_b = q_b + x * q_a;
p_a = new_p_a ;
q_a = new_q_a;
p_b = new_p_b ;
q_b = new_q_b;
/* compute the number of steps to the right */
x_num = alpha_num * q_b - denum * p_b;
x_denum = - alpha_num * q_a + denum * p_a;
x = (x_num + x_denum - 1) / x_denum; /* x = ceil(x_num / x_denum) */
/* check whether we have a valid approximation */
aa = matches(p_b + x * p_a, q_b + x * q_a, alpha_num, d_num, denum);
bb = matches(p_b + (x - 1) * p_a, q_b + (x - 1) * q_a, alpha_num, d_num, denum);
if (aa || bb) {
find_exact_solution_right(p_a, q_a, p_b, q_b, alpha_num, d_num, denum, p, q);
return 0;
}
/* update the interval */
new_p_a = p_b + (x - 1) * p_a;
new_q_a = q_b + (x - 1) * q_a;
new_p_b = p_b + x * p_a;
new_q_b = q_b + x * q_a;
p_a = new_p_a;
q_a = new_q_a;
p_b = new_p_b;
q_b = new_q_b;
}
}
int rte_approx(double alpha, double d, uint32_t *p, uint32_t *q)
{
uint32_t alpha_num, d_num, denum;
/* Check input arguments */
if (!((0.0 < d) && (d < alpha) && (alpha < 1.0))) {
return -1;
}
if ((p == NULL) || (q == NULL)) {
return -2;
}
/* Compute alpha_num, d_num and denum */
denum = 1;
while (d < 1) {
alpha *= 10;
d *= 10;
denum *= 10;
}
alpha_num = (uint32_t) alpha;
d_num = (uint32_t) d;
/* Perform approximation */
return find_best_rational_approximation(alpha_num, d_num, denum, p, q);
}
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