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geom_predicates.c
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/*****************************************************************************/
/* */
/* Routines for Arbitrary Precision Floating-point Arithmetic */
/* and Fast Robust Geometric Predicates */
/* (predicates.c) */
/* */
/* May 18, 1996 */
/* */
/* Placed in the public domain by */
/* Jonathan Richard Shewchuk */
/* School of Computer Science */
/* Carnegie Mellon University */
/* 5000 Forbes Avenue */
/* Pittsburgh, Pennsylvania 15213-3891 */
/* [email protected] */
/* */
/* This file contains C implementation of algorithms for exact addition */
/* and multiplication of floating-point numbers, and predicates for */
/* robustly performing the orientation and incircle tests used in */
/* computational geometry. The algorithms and underlying theory are */
/* described in Jonathan Richard Shewchuk. "Adaptive Precision Floating- */
/* Point Arithmetic and Fast Robust Geometric Predicates." Technical */
/* Report CMU-CS-96-140, School of Computer Science, Carnegie Mellon */
/* University, Pittsburgh, Pennsylvania, May 1996. (Submitted to */
/* Discrete & Computational Geometry.) */
/* */
/* This file, the paper listed above, and other information are available */
/* from the Web page http://www.cs.cmu.edu/~quake/robust.html . */
/* */
/*****************************************************************************/
/*****************************************************************************/
/* */
/* Using this code: */
/* */
/* First, read the short or long version of the paper (from the Web page */
/* above). */
/* */
/* Be sure to call exactinit() once, before calling any of the arithmetic */
/* functions or geometric predicates. Also be sure to turn on the */
/* optimizer when compiling this file. */
/* */
/* */
/* Several geometric predicates are defined. Their parameters are all */
/* points. Each point is an array of two or three floating-point */
/* numbers. The geometric predicates, described in the papers, are */
/* */
/* orient2d(pa, pb, pc) */
/* orient2dfast(pa, pb, pc) */
/* orient3d(pa, pb, pc, pd) */
/* orient3dfast(pa, pb, pc, pd) */
/* incircle(pa, pb, pc, pd) */
/* incirclefast(pa, pb, pc, pd) */
/* insphere(pa, pb, pc, pd, pe) */
/* inspherefast(pa, pb, pc, pd, pe) */
/* */
/* Those with suffix "fast" are approximate, non-robust versions. Those */
/* without the suffix are adaptive precision, robust versions. There */
/* are also versions with the suffices "exact" and "slow", which are */
/* non-adaptive, exact arithmetic versions, which I use only for timings */
/* in my arithmetic papers. */
/* */
/* */
/* An expansion is represented by an array of floating-point numbers, */
/* sorted from smallest to largest magnitude (possibly with interspersed */
/* zeros). The length of each expansion is stored as a separate integer, */
/* and each arithmetic function returns an integer which is the length */
/* of the expansion it created. */
/* */
/* Several arithmetic functions are defined. Their parameters are */
/* */
/* e, f Input expansions */
/* elen, flen Lengths of input expansions (must be >= 1) */
/* h Output expansion */
/* b Input scalar */
/* */
/* The arithmetic functions are */
/* */
/* grow_expansion(elen, e, b, h) */
/* grow_expansion_zeroelim(elen, e, b, h) */
/* expansion_sum(elen, e, flen, f, h) */
/* expansion_sum_zeroelim1(elen, e, flen, f, h) */
/* expansion_sum_zeroelim2(elen, e, flen, f, h) */
/* fast_expansion_sum(elen, e, flen, f, h) */
/* fast_expansion_sum_zeroelim(elen, e, flen, f, h) */
/* linear_expansion_sum(elen, e, flen, f, h) */
/* linear_expansion_sum_zeroelim(elen, e, flen, f, h) */
/* scale_expansion(elen, e, b, h) */
/* scale_expansion_zeroelim(elen, e, b, h) */
/* compress(elen, e, h) */
/* */
/* All of these are described in the long version of the paper; some are */
/* described in the short version. All return an integer that is the */
/* length of h. Those with suffix _zeroelim perform zero elimination, */
/* and are recommended over their counterparts. The procedure */
/* fast_expansion_sum_zeroelim() (or linear_expansion_sum_zeroelim() on */
/* processors that do not use the round-to-even tiebreaking rule) is */
/* recommended over expansion_sum_zeroelim(). Each procedure has a */
/* little note next to it (in the code below) that tells you whether or */
/* not the output expansion may be the same array as one of the input */
/* expansions. */
/* */
/* */
/* If you look around below, you'll also find macros for a bunch of */
/* simple unrolled arithmetic operations, and procedures for printing */
/* expansions (commented out because they don't work with all C */
/* compilers) and for generating random floating-point numbers whose */
/* significand bits are all random. Most of the macros have undocumented */
/* requirements that certain of their parameters should not be the same */
/* variable; for safety, better to make sure all the parameters are */
/* distinct variables. Feel free to send email to [email protected] if you */
/* have questions. */
/* */
/*****************************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <sys/time.h>
/* On some machines, the exact arithmetic routines might be defeated by the */
/* use of internal extended precision floating-point registers. Sometimes */
/* this problem can be fixed by defining certain values to be volatile, */
/* thus forcing them to be stored to memory and rounded off. This isn't */
/* a great solution, though, as it slows the arithmetic down. */
/* */
/* To try this out, write "#define INEXACT volatile" below. Normally, */
/* however, INEXACT should be defined to be nothing. ("#define INEXACT".) */
#define INEXACT /* Nothing */
/* #define INEXACT volatile */
#define random() rand()
#define REAL double /* float or double */
#define REALPRINT doubleprint
#define REALRAND doublerand
#define NARROWRAND narrowdoublerand
#define UNIFORMRAND uniformdoublerand
/* Which of the following two methods of finding the absolute values is */
/* fastest is compiler-dependent. A few compilers can inline and optimize */
/* the fabs() call; but most will incur the overhead of a function call, */
/* which is disastrously slow. A faster way on IEEE machines might be to */
/* mask the appropriate bit, but that's difficult to do in C. */
#define Absolute(a) ((a) >= 0.0 ? (a) : -(a))
/* #define Absolute(a) fabs(a) */
/* Many of the operations are broken up into two pieces, a main part that */
/* performs an approximate operation, and a "tail" that computes the */
/* roundoff error of that operation. */
/* */
/* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */
/* Split(), and Two_Product() are all implemented as described in the */
/* reference. Each of these macros requires certain variables to be */
/* defined in the calling routine. The variables `bvirt', `c', `abig', */
/* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */
/* they store the result of an operation that may incur roundoff error. */
/* The input parameter `x' (or the highest numbered `x_' parameter) must */
/* also be declared `INEXACT'. */
#define Fast_Two_Sum_Tail(a, b, x, y) \
bvirt = x - a; \
y = b - bvirt
#define Fast_Two_Sum(a, b, x, y) \
x = (REAL) (a + b); \
Fast_Two_Sum_Tail(a, b, x, y)
#define Fast_Two_Diff_Tail(a, b, x, y) \
bvirt = a - x; \
y = bvirt - b
#define Fast_Two_Diff(a, b, x, y) \
x = (REAL) (a - b); \
Fast_Two_Diff_Tail(a, b, x, y)
#define Two_Sum_Tail(a, b, x, y) \
bvirt = (REAL) (x - a); \
avirt = x - bvirt; \
bround = b - bvirt; \
around = a - avirt; \
y = around + bround
#define Two_Sum(a, b, x, y) \
x = (REAL) (a + b); \
Two_Sum_Tail(a, b, x, y)
#define Two_Diff_Tail(a, b, x, y) \
bvirt = (REAL) (a - x); \
avirt = x + bvirt; \
bround = bvirt - b; \
around = a - avirt; \
y = around + bround
#define Two_Diff(a, b, x, y) \
x = (REAL) (a - b); \
Two_Diff_Tail(a, b, x, y)
#define Split(a, ahi, alo) \
c = (REAL) (splitter * a); \
abig = (REAL) (c - a); \
ahi = c - abig; \
alo = a - ahi
#define Two_Product_Tail(a, b, x, y) \
Split(a, ahi, alo); \
Split(b, bhi, blo); \
err1 = x - (ahi * bhi); \
err2 = err1 - (alo * bhi); \
err3 = err2 - (ahi * blo); \
y = (alo * blo) - err3
#define Two_Product(a, b, x, y) \
x = (REAL) (a * b); \
Two_Product_Tail(a, b, x, y)
/* Two_Product_Presplit() is Two_Product() where one of the inputs has */
/* already been split. Avoids redundant splitting. */
#define Two_Product_Presplit(a, b, bhi, blo, x, y) \
x = (REAL) (a * b); \
Split(a, ahi, alo); \
err1 = x - (ahi * bhi); \
err2 = err1 - (alo * bhi); \
err3 = err2 - (ahi * blo); \
y = (alo * blo) - err3
/* Two_Product_2Presplit() is Two_Product() where both of the inputs have */
/* already been split. Avoids redundant splitting. */
#define Two_Product_2Presplit(a, ahi, alo, b, bhi, blo, x, y) \
x = (REAL) (a * b); \
err1 = x - (ahi * bhi); \
err2 = err1 - (alo * bhi); \
err3 = err2 - (ahi * blo); \
y = (alo * blo) - err3
/* Square() can be done more quickly than Two_Product(). */
#define Square_Tail(a, x, y) \
Split(a, ahi, alo); \
err1 = x - (ahi * ahi); \
err3 = err1 - ((ahi + ahi) * alo); \
y = (alo * alo) - err3
#define Square(a, x, y) \
x = (REAL) (a * a); \
Square_Tail(a, x, y)
/* Macros for summing expansions of various fixed lengths. These are all */
/* unrolled versions of Expansion_Sum(). */
#define Two_One_Sum(a1, a0, b, x2, x1, x0) \
Two_Sum(a0, b , _i, x0); \
Two_Sum(a1, _i, x2, x1)
#define Two_One_Diff(a1, a0, b, x2, x1, x0) \
Two_Diff(a0, b , _i, x0); \
Two_Sum( a1, _i, x2, x1)
#define Two_Two_Sum(a1, a0, b1, b0, x3, x2, x1, x0) \
Two_One_Sum(a1, a0, b0, _j, _0, x0); \
Two_One_Sum(_j, _0, b1, x3, x2, x1)
#define Two_Two_Diff(a1, a0, b1, b0, x3, x2, x1, x0) \
Two_One_Diff(a1, a0, b0, _j, _0, x0); \
Two_One_Diff(_j, _0, b1, x3, x2, x1)
#define Four_One_Sum(a3, a2, a1, a0, b, x4, x3, x2, x1, x0) \
Two_One_Sum(a1, a0, b , _j, x1, x0); \
Two_One_Sum(a3, a2, _j, x4, x3, x2)
#define Four_Two_Sum(a3, a2, a1, a0, b1, b0, x5, x4, x3, x2, x1, x0) \
Four_One_Sum(a3, a2, a1, a0, b0, _k, _2, _1, _0, x0); \
Four_One_Sum(_k, _2, _1, _0, b1, x5, x4, x3, x2, x1)
#define Four_Four_Sum(a3, a2, a1, a0, b4, b3, b1, b0, x7, x6, x5, x4, x3, x2, \
x1, x0) \
Four_Two_Sum(a3, a2, a1, a0, b1, b0, _l, _2, _1, _0, x1, x0); \
Four_Two_Sum(_l, _2, _1, _0, b4, b3, x7, x6, x5, x4, x3, x2)
#define Eight_One_Sum(a7, a6, a5, a4, a3, a2, a1, a0, b, x8, x7, x6, x5, x4, \
x3, x2, x1, x0) \
Four_One_Sum(a3, a2, a1, a0, b , _j, x3, x2, x1, x0); \
Four_One_Sum(a7, a6, a5, a4, _j, x8, x7, x6, x5, x4)
#define Eight_Two_Sum(a7, a6, a5, a4, a3, a2, a1, a0, b1, b0, x9, x8, x7, \
x6, x5, x4, x3, x2, x1, x0) \
Eight_One_Sum(a7, a6, a5, a4, a3, a2, a1, a0, b0, _k, _6, _5, _4, _3, _2, \
_1, _0, x0); \
Eight_One_Sum(_k, _6, _5, _4, _3, _2, _1, _0, b1, x9, x8, x7, x6, x5, x4, \
x3, x2, x1)
#define Eight_Four_Sum(a7, a6, a5, a4, a3, a2, a1, a0, b4, b3, b1, b0, x11, \
x10, x9, x8, x7, x6, x5, x4, x3, x2, x1, x0) \
Eight_Two_Sum(a7, a6, a5, a4, a3, a2, a1, a0, b1, b0, _l, _6, _5, _4, _3, \
_2, _1, _0, x1, x0); \
Eight_Two_Sum(_l, _6, _5, _4, _3, _2, _1, _0, b4, b3, x11, x10, x9, x8, \
x7, x6, x5, x4, x3, x2)
/* Macros for multiplying expansions of various fixed lengths. */
#define Two_One_Product(a1, a0, b, x3, x2, x1, x0) \
Split(b, bhi, blo); \
Two_Product_Presplit(a0, b, bhi, blo, _i, x0); \
Two_Product_Presplit(a1, b, bhi, blo, _j, _0); \
Two_Sum(_i, _0, _k, x1); \
Fast_Two_Sum(_j, _k, x3, x2)
#define Four_One_Product(a3, a2, a1, a0, b, x7, x6, x5, x4, x3, x2, x1, x0) \
Split(b, bhi, blo); \
Two_Product_Presplit(a0, b, bhi, blo, _i, x0); \
Two_Product_Presplit(a1, b, bhi, blo, _j, _0); \
Two_Sum(_i, _0, _k, x1); \
Fast_Two_Sum(_j, _k, _i, x2); \
Two_Product_Presplit(a2, b, bhi, blo, _j, _0); \
Two_Sum(_i, _0, _k, x3); \
Fast_Two_Sum(_j, _k, _i, x4); \
Two_Product_Presplit(a3, b, bhi, blo, _j, _0); \
Two_Sum(_i, _0, _k, x5); \
Fast_Two_Sum(_j, _k, x7, x6)
#define Two_Two_Product(a1, a0, b1, b0, x7, x6, x5, x4, x3, x2, x1, x0) \
Split(a0, a0hi, a0lo); \
Split(b0, bhi, blo); \
Two_Product_2Presplit(a0, a0hi, a0lo, b0, bhi, blo, _i, x0); \
Split(a1, a1hi, a1lo); \
Two_Product_2Presplit(a1, a1hi, a1lo, b0, bhi, blo, _j, _0); \
Two_Sum(_i, _0, _k, _1); \
Fast_Two_Sum(_j, _k, _l, _2); \
Split(b1, bhi, blo); \
Two_Product_2Presplit(a0, a0hi, a0lo, b1, bhi, blo, _i, _0); \
Two_Sum(_1, _0, _k, x1); \
Two_Sum(_2, _k, _j, _1); \
Two_Sum(_l, _j, _m, _2); \
Two_Product_2Presplit(a1, a1hi, a1lo, b1, bhi, blo, _j, _0); \
Two_Sum(_i, _0, _n, _0); \
Two_Sum(_1, _0, _i, x2); \
Two_Sum(_2, _i, _k, _1); \
Two_Sum(_m, _k, _l, _2); \
Two_Sum(_j, _n, _k, _0); \
Two_Sum(_1, _0, _j, x3); \
Two_Sum(_2, _j, _i, _1); \
Two_Sum(_l, _i, _m, _2); \
Two_Sum(_1, _k, _i, x4); \
Two_Sum(_2, _i, _k, x5); \
Two_Sum(_m, _k, x7, x6)
/* An expansion of length two can be squared more quickly than finding the */
/* product of two different expansions of length two, and the result is */
/* guaranteed to have no more than six (rather than eight) components. */
#define Two_Square(a1, a0, x5, x4, x3, x2, x1, x0) \
Square(a0, _j, x0); \
_0 = a0 + a0; \
Two_Product(a1, _0, _k, _1); \
Two_One_Sum(_k, _1, _j, _l, _2, x1); \
Square(a1, _j, _1); \
Two_Two_Sum(_j, _1, _l, _2, x5, x4, x3, x2)
REAL splitter; /* = 2^ceiling(p / 2) + 1. Used to split floats in half. */
REAL epsilon; /* = 2^(-p). Used to estimate roundoff errors. */
/* A set of coefficients used to calculate maximum roundoff errors. */
REAL resulterrbound;
REAL ccwerrboundA, ccwerrboundB, ccwerrboundC;
REAL o3derrboundA, o3derrboundB, o3derrboundC;
REAL iccerrboundA, iccerrboundB, iccerrboundC;
REAL isperrboundA, isperrboundB, isperrboundC;
/*****************************************************************************/
/* */
/* doubleprint() Print the bit representation of a double. */
/* */
/* Useful for debugging exact arithmetic routines. */
/* */
/*****************************************************************************/
/*
static void doubleprint(number)
double number;
{
unsigned long long no;
unsigned long long sign, expo;
int exponent;
int i, bottomi;
no = *(unsigned long long *) &number;
sign = no & 0x8000000000000000ll;
expo = (no >> 52) & 0x7ffll;
exponent = (int) expo;
exponent = exponent - 1023;
if (sign) {
printf("-");
} else {
printf(" ");
}
if (exponent == -1023) {
printf(
"0.0000000000000000000000000000000000000000000000000000_ ( )");
} else {
printf("1.");
bottomi = -1;
for (i = 0; i < 52; i++) {
if (no & 0x0008000000000000ll) {
printf("1");
bottomi = i;
} else {
printf("0");
}
no <<= 1;
}
printf("_%d (%d)", exponent, exponent - 1 - bottomi);
}
}
*/
/*****************************************************************************/
/* */
/* floatprint() Print the bit representation of a float. */
/* */
/* Useful for debugging exact arithmetic routines. */
/* */
/*****************************************************************************/
/*
static void floatprint(number)
float number;
{
unsigned no;
unsigned sign, expo;
int exponent;
int i, bottomi;
no = *(unsigned *) &number;
sign = no & 0x80000000;
expo = (no >> 23) & 0xff;
exponent = (int) expo;
exponent = exponent - 127;
if (sign) {
printf("-");
} else {
printf(" ");
}
if (exponent == -127) {
printf("0.00000000000000000000000_ ( )");
} else {
printf("1.");
bottomi = -1;
for (i = 0; i < 23; i++) {
if (no & 0x00400000) {
printf("1");
bottomi = i;
} else {
printf("0");
}
no <<= 1;
}
printf("_%3d (%3d)", exponent, exponent - 1 - bottomi);
}
}
*/
/*****************************************************************************/
/* */
/* expansion_print() Print the bit representation of an expansion. */
/* */
/* Useful for debugging exact arithmetic routines. */
/* */
/*****************************************************************************/
/*
static void expansion_print(elen, e)
int elen;
REAL *e;
{
int i;
for (i = elen - 1; i >= 0; i--) {
REALPRINT(e[i]);
if (i > 0) {
printf(" +\n");
} else {
printf("\n");
}
}
}
*/
/*****************************************************************************/
/* */
/* doublerand() Generate a double with random 53-bit significand and a */
/* random exponent in [0, 511]. */
/* */
/*****************************************************************************/
static double doublerand()
{
double result;
double expo;
long a, b, c;
long i;
a = random();
b = random();
c = random();
result = (double) (a - 1073741824) * 8388608.0 + (double) (b >> 8);
for (i = 512, expo = 2; i <= 131072; i *= 2, expo = expo * expo) {
if (c & i) {
result *= expo;
}
}
return result;
}
/*****************************************************************************/
/* */
/* narrowdoublerand() Generate a double with random 53-bit significand */
/* and a random exponent in [0, 7]. */
/* */
/*****************************************************************************/
static double narrowdoublerand()
{
double result;
double expo;
long a, b, c;
long i;
a = random();
b = random();
c = random();
result = (double) (a - 1073741824) * 8388608.0 + (double) (b >> 8);
for (i = 512, expo = 2; i <= 2048; i *= 2, expo = expo * expo) {
if (c & i) {
result *= expo;
}
}
return result;
}
/*****************************************************************************/
/* */
/* uniformdoublerand() Generate a double with random 53-bit significand. */
/* */
/*****************************************************************************/
double geom_rand_unifd(){
double result;
long a, b;
a = random();
b = random();
result = (double) (a - 1073741824) * 8388608.0 + (double) (b >> 8);
return result;
}
/*****************************************************************************/
/* */
/* floatrand() Generate a float with random 24-bit significand and a */
/* random exponent in [0, 63]. */
/* */
/*****************************************************************************/
static float floatrand()
{
float result;
float expo;
long a, c;
long i;
a = random();
c = random();
result = (float) ((a - 1073741824) >> 6);
for (i = 512, expo = 2; i <= 16384; i *= 2, expo = expo * expo) {
if (c & i) {
result *= expo;
}
}
return result;
}
/*****************************************************************************/
/* */
/* narrowfloatrand() Generate a float with random 24-bit significand and */
/* a random exponent in [0, 7]. */
/* */
/*****************************************************************************/
static float narrowfloatrand()
{
float result;
float expo;
long a, c;
long i;
a = random();
c = random();
result = (float) ((a - 1073741824) >> 6);
for (i = 512, expo = 2; i <= 2048; i *= 2, expo = expo * expo) {
if (c & i) {
result *= expo;
}
}
return result;
}
/*****************************************************************************/
/* */
/* uniformfloatrand() Generate a float with random 24-bit significand. */
/* */
/*****************************************************************************/
float geom_rand_uniff(){
float result;
long a;
a = random();
result = (float) ((a - 1073741824) >> 6);
return result;
}
/*****************************************************************************/
/* */
/* exactinit() Initialize the variables used for exact arithmetic. */
/* */
/* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */
/* floating-point arithmetic. `epsilon' bounds the relative roundoff */
/* error. It is used for floating-point error analysis. */
/* */
/* `splitter' is used to split floating-point numbers into two half- */
/* length significands for exact multiplication. */
/* */
/* I imagine that a highly optimizing compiler might be too smart for its */
/* own good, and somehow cause this routine to fail, if it pretends that */
/* floating-point arithmetic is too much like real arithmetic. */
/* */
/* Don't change this routine unless you fully understand it. */
/* */
/*****************************************************************************/
void geom_predicates_init()
{
REAL half;
REAL check, lastcheck;
int every_other;
every_other = 1;
half = 0.5;
epsilon = 1.0;
splitter = 1.0;
check = 1.0;
/* Repeatedly divide `epsilon' by two until it is too small to add to */
/* one without causing roundoff. (Also check if the sum is equal to */
/* the previous sum, for machines that round up instead of using exact */
/* rounding. Not that this library will work on such machines anyway. */
do {
lastcheck = check;
epsilon *= half;
if (every_other) {
splitter *= 2.0;
}
every_other = !every_other;
check = 1.0 + epsilon;
} while ((check != 1.0) && (check != lastcheck));
splitter += 1.0;
/* Error bounds for orientation and incircle tests. */
resulterrbound = (3.0 + 8.0 * epsilon) * epsilon;
ccwerrboundA = (3.0 + 16.0 * epsilon) * epsilon;
ccwerrboundB = (2.0 + 12.0 * epsilon) * epsilon;
ccwerrboundC = (9.0 + 64.0 * epsilon) * epsilon * epsilon;
o3derrboundA = (7.0 + 56.0 * epsilon) * epsilon;
o3derrboundB = (3.0 + 28.0 * epsilon) * epsilon;
o3derrboundC = (26.0 + 288.0 * epsilon) * epsilon * epsilon;
iccerrboundA = (10.0 + 96.0 * epsilon) * epsilon;
iccerrboundB = (4.0 + 48.0 * epsilon) * epsilon;
iccerrboundC = (44.0 + 576.0 * epsilon) * epsilon * epsilon;
isperrboundA = (16.0 + 224.0 * epsilon) * epsilon;
isperrboundB = (5.0 + 72.0 * epsilon) * epsilon;
isperrboundC = (71.0 + 1408.0 * epsilon) * epsilon * epsilon;
}
/*****************************************************************************/
/* */
/* grow_expansion() Add a scalar to an expansion. */
/* */
/* Sets h = e + b. See the long version of my paper for details. */
/* */
/* Maintains the nonoverlapping property. If round-to-even is used (as */
/* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */
/* properties as well. (That is, if e has one of these properties, so */
/* will h.) */
/* */
/*****************************************************************************/
static int grow_expansion(elen, e, b, h) /* e and h can be the same. */
int elen;
REAL *e;
REAL b;
REAL *h;
{
REAL Q;
INEXACT REAL Qnew;
int eindex;
REAL enow;
INEXACT REAL bvirt;
REAL avirt, bround, around;
Q = b;
for (eindex = 0; eindex < elen; eindex++) {
enow = e[eindex];
Two_Sum(Q, enow, Qnew, h[eindex]);
Q = Qnew;
}
h[eindex] = Q;
return eindex + 1;
}
/*****************************************************************************/
/* */
/* grow_expansion_zeroelim() Add a scalar to an expansion, eliminating */
/* zero components from the output expansion. */
/* */
/* Sets h = e + b. See the long version of my paper for details. */
/* */
/* Maintains the nonoverlapping property. If round-to-even is used (as */
/* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */
/* properties as well. (That is, if e has one of these properties, so */
/* will h.) */
/* */
/*****************************************************************************/
static int grow_expansion_zeroelim(elen, e, b, h) /* e and h can be the same. */
int elen;
REAL *e;
REAL b;
REAL *h;
{
REAL Q, hh;
INEXACT REAL Qnew;
int eindex, hindex;
REAL enow;
INEXACT REAL bvirt;
REAL avirt, bround, around;
hindex = 0;
Q = b;
for (eindex = 0; eindex < elen; eindex++) {
enow = e[eindex];
Two_Sum(Q, enow, Qnew, hh);
Q = Qnew;
if (hh != 0.0) {
h[hindex++] = hh;
}
}
if ((Q != 0.0) || (hindex == 0)) {
h[hindex++] = Q;
}
return hindex;
}
/*****************************************************************************/
/* */
/* expansion_sum() Sum two expansions. */
/* */
/* Sets h = e + f. See the long version of my paper for details. */
/* */
/* Maintains the nonoverlapping property. If round-to-even is used (as */
/* with IEEE 754), maintains the nonadjacent property as well. (That is, */
/* if e has one of these properties, so will h.) Does NOT maintain the */
/* strongly nonoverlapping property. */
/* */
/*****************************************************************************/
static int expansion_sum(elen, e, flen, f, h)
/* e and h can be the same, but f and h cannot. */
int elen;
REAL *e;
int flen;
REAL *f;
REAL *h;
{
REAL Q;
INEXACT REAL Qnew;
int findex, hindex, hlast;
REAL hnow;
INEXACT REAL bvirt;
REAL avirt, bround, around;
Q = f[0];
for (hindex = 0; hindex < elen; hindex++) {
hnow = e[hindex];
Two_Sum(Q, hnow, Qnew, h[hindex]);
Q = Qnew;
}
h[hindex] = Q;
hlast = hindex;
for (findex = 1; findex < flen; findex++) {
Q = f[findex];
for (hindex = findex; hindex <= hlast; hindex++) {
hnow = h[hindex];
Two_Sum(Q, hnow, Qnew, h[hindex]);
Q = Qnew;
}
h[++hlast] = Q;
}
return hlast + 1;
}
/*****************************************************************************/
/* */
/* expansion_sum_zeroelim1() Sum two expansions, eliminating zero */
/* components from the output expansion. */
/* */
/* Sets h = e + f. See the long version of my paper for details. */
/* */
/* Maintains the nonoverlapping property. If round-to-even is used (as */
/* with IEEE 754), maintains the nonadjacent property as well. (That is, */
/* if e has one of these properties, so will h.) Does NOT maintain the */
/* strongly nonoverlapping property. */
/* */
/*****************************************************************************/
static int expansion_sum_zeroelim1(elen, e, flen, f, h)
/* e and h can be the same, but f and h cannot. */
int elen;
REAL *e;
int flen;
REAL *f;
REAL *h;
{
REAL Q;
INEXACT REAL Qnew;
int index, findex, hindex, hlast;
REAL hnow;
INEXACT REAL bvirt;
REAL avirt, bround, around;
Q = f[0];
for (hindex = 0; hindex < elen; hindex++) {
hnow = e[hindex];
Two_Sum(Q, hnow, Qnew, h[hindex]);
Q = Qnew;
}
h[hindex] = Q;
hlast = hindex;
for (findex = 1; findex < flen; findex++) {
Q = f[findex];
for (hindex = findex; hindex <= hlast; hindex++) {
hnow = h[hindex];
Two_Sum(Q, hnow, Qnew, h[hindex]);
Q = Qnew;
}
h[++hlast] = Q;
}
hindex = -1;
for (index = 0; index <= hlast; index++) {
hnow = h[index];
if (hnow != 0.0) {
h[++hindex] = hnow;
}
}
if (hindex == -1) {
return 1;
} else {
return hindex + 1;
}
}
/*****************************************************************************/
/* */
/* expansion_sum_zeroelim2() Sum two expansions, eliminating zero */
/* components from the output expansion. */
/* */
/* Sets h = e + f. See the long version of my paper for details. */
/* */
/* Maintains the nonoverlapping property. If round-to-even is used (as */
/* with IEEE 754), maintains the nonadjacent property as well. (That is, */
/* if e has one of these properties, so will h.) Does NOT maintain the */
/* strongly nonoverlapping property. */
/* */
/*****************************************************************************/
static int expansion_sum_zeroelim2(elen, e, flen, f, h)
/* e and h can be the same, but f and h cannot. */
int elen;
REAL *e;
int flen;
REAL *f;
REAL *h;
{
REAL Q, hh;
INEXACT REAL Qnew;
int eindex, findex, hindex, hlast;
REAL enow;
INEXACT REAL bvirt;
REAL avirt, bround, around;
hindex = 0;
Q = f[0];
for (eindex = 0; eindex < elen; eindex++) {
enow = e[eindex];
Two_Sum(Q, enow, Qnew, hh);
Q = Qnew;
if (hh != 0.0) {
h[hindex++] = hh;
}
}
h[hindex] = Q;
hlast = hindex;
for (findex = 1; findex < flen; findex++) {
hindex = 0;
Q = f[findex];
for (eindex = 0; eindex <= hlast; eindex++) {
enow = h[eindex];
Two_Sum(Q, enow, Qnew, hh);
Q = Qnew;
if (hh != 0) {
h[hindex++] = hh;
}
}
h[hindex] = Q;
hlast = hindex;
}
return hlast + 1;
}
/*****************************************************************************/
/* */
/* fast_expansion_sum() Sum two expansions. */
/* */
/* Sets h = e + f. See the long version of my paper for details. */
/* */
/* If round-to-even is used (as with IEEE 754), maintains the strongly */
/* nonoverlapping property. (That is, if e is strongly nonoverlapping, h */
/* will be also.) Does NOT maintain the nonoverlapping or nonadjacent */
/* properties. */
/* */
/*****************************************************************************/
static int fast_expansion_sum(elen, e, flen, f, h) /* h cannot be e or f. */
int elen;
REAL *e;
int flen;
REAL *f;
REAL *h;
{
REAL Q;
INEXACT REAL Qnew;
INEXACT REAL bvirt;
REAL avirt, bround, around;
int eindex, findex, hindex;
REAL enow, fnow;
enow = e[0];
fnow = f[0];
eindex = findex = 0;
if ((fnow > enow) == (fnow > -enow)) {
Q = enow;
enow = e[++eindex];
} else {
Q = fnow;
fnow = f[++findex];
}
hindex = 0;
if ((eindex < elen) && (findex < flen)) {
if ((fnow > enow) == (fnow > -enow)) {
Fast_Two_Sum(enow, Q, Qnew, h[0]);
enow = e[++eindex];
} else {
Fast_Two_Sum(fnow, Q, Qnew, h[0]);
fnow = f[++findex];
}
Q = Qnew;
hindex = 1;
while ((eindex < elen) && (findex < flen)) {
if ((fnow > enow) == (fnow > -enow)) {
Two_Sum(Q, enow, Qnew, h[hindex]);
enow = e[++eindex];
} else {
Two_Sum(Q, fnow, Qnew, h[hindex]);
fnow = f[++findex];
}
Q = Qnew;
hindex++;
}
}
while (eindex < elen) {
Two_Sum(Q, enow, Qnew, h[hindex]);
enow = e[++eindex];
Q = Qnew;
hindex++;
}
while (findex < flen) {
Two_Sum(Q, fnow, Qnew, h[hindex]);
fnow = f[++findex];
Q = Qnew;
hindex++;