/*****************************************************************************/ /* */ /* 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 */ /* jrs@cs.cmu.edu */ /* */ /* 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 jrs@cs.cmu.edu if you */ /* have questions. */ /* */ /*****************************************************************************/ #include #include #include #include #include "predicates.h" /* FPU control. We MUST have only double precision (not extended precision) */ #include "rounding.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 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) /* 2^(-p), where p=DBL_MANT_DIG. Used to estimate roundoff errors. */ static const REAL epsilon=0.5*DBL_EPSILON; /* 2^ceiling(p/2) + 1. Used to split floats in half. */ static const REAL splitter=sqrt((DBL_MANT_DIG % 2 ? 2.0 : 1.0)/epsilon)+1.0; /* A set of coefficients used to calculate maximum roundoff errors. */ const REAL resulterrbound=(3.0 + 8.0 * epsilon) * epsilon; const REAL ccwerrboundA=(3.0 + 16.0 * epsilon) * epsilon; const REAL ccwerrboundB=(2.0 + 12.0 * epsilon) * epsilon; const REAL ccwerrboundC=(9.0 + 64.0 * epsilon) * epsilon * epsilon; const REAL o3derrboundA=(7.0 + 56.0 * epsilon) * epsilon; const REAL o3derrboundB=(3.0 + 28.0 * epsilon) * epsilon; const REAL o3derrboundC=(26.0 + 288.0 * epsilon) * epsilon * epsilon; const REAL iccerrboundA=(10.0 + 96.0 * epsilon) * epsilon; const REAL iccerrboundB=(4.0 + 48.0 * epsilon) * epsilon; const REAL iccerrboundC=(44.0 + 576.0 * epsilon) * epsilon * epsilon; const REAL isperrboundA=(16.0 + 224.0 * epsilon) * epsilon; const REAL isperrboundB=(5.0 + 72.0 * epsilon) * epsilon; const REAL isperrboundC=(71.0 + 1408.0 * epsilon) * epsilon * epsilon; /*****************************************************************************/ /* */ /* doubleprint() Print the bit representation of a double. */ /* */ /* Useful for debugging exact arithmetic routines. */ /* */ /*****************************************************************************/ /* 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. */ /* */ /*****************************************************************************/ /* 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. */ /* */ /*****************************************************************************/ /* 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. */ /* */ /*****************************************************************************/ /* static double uniformdoublerand() { 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. */ /* */ /*****************************************************************************/ /* static float uniformfloatrand() { float result; long a; a = random(); result = (float) ((a - 1073741824) >> 6); return result; } */ /*****************************************************************************/ /* */ /* fast_expansion_sum_zeroelim() Sum two expansions, eliminating zero */ /* components from the output expansion. */ /* */ /* 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_zeroelim(int elen, const REAL *e, int flen, const REAL *f, REAL *h) /* h cannot be e or f. */ { REAL Q; INEXACT REAL Qnew; INEXACT REAL hh; 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, hh); enow = e[++eindex]; } else { Fast_Two_Sum(fnow, Q, Qnew, hh); fnow = f[++findex]; } Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } while ((eindex < elen) && (findex < flen)) { if ((fnow > enow) == (fnow > -enow)) { Two_Sum(Q, enow, Qnew, hh); enow = e[++eindex]; } else { Two_Sum(Q, fnow, Qnew, hh); fnow = f[++findex]; } Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } } } while (eindex < elen) { Two_Sum(Q, enow, Qnew, hh); enow = e[++eindex]; Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } } while (findex < flen) { Two_Sum(Q, fnow, Qnew, hh); fnow = f[++findex]; Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } } if ((Q != 0.0) || (hindex == 0)) { h[hindex++] = Q; } return hindex; } /*****************************************************************************/ /* */ /* scale_expansion_zeroelim() Multiply an expansion by a scalar, */ /* eliminating zero components from the */ /* output expansion. */ /* */ /* Sets h = be. See either 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 scale_expansion_zeroelim(int elen, const REAL *e, REAL b, REAL *h) /* e and h cannot be the same. */ { INEXACT REAL Q, sum; REAL hh; INEXACT REAL product1; REAL product0; int eindex, hindex; REAL enow; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; Split(b, bhi, blo); Two_Product_Presplit(e[0], b, bhi, blo, Q, hh); hindex = 0; if (hh != 0) { h[hindex++] = hh; } for (eindex = 1; eindex < elen; eindex++) { enow = e[eindex]; Two_Product_Presplit(enow, b, bhi, blo, product1, product0); Two_Sum(Q, product0, sum, hh); if (hh != 0) { h[hindex++] = hh; } Fast_Two_Sum(product1, sum, Q, hh); if (hh != 0) { h[hindex++] = hh; } } if ((Q != 0.0) || (hindex == 0)) { h[hindex++] = Q; } return hindex; } /*****************************************************************************/ /* */ /* estimate() Produce a one-word estimate of an expansion's value. */ /* */ /* See either version of my paper for details. */ /* */ /*****************************************************************************/ static REAL estimate(int elen, const REAL *e) { REAL Q; int eindex; Q = e[0]; for (eindex = 1; eindex < elen; eindex++) { Q += e[eindex]; } return Q; } /*****************************************************************************/ /* */ /* orient2dfast() Approximate 2D orientation test. Nonrobust. */ /* orient2dexact() Exact 2D orientation test. Robust. */ /* orient2dslow() Another exact 2D orientation test. Robust. */ /* orient2d() Adaptive exact 2D orientation test. Robust. */ /* */ /* Return a positive value if the points pa, pb, and pc occur */ /* in counterclockwise order; a negative value if they occur */ /* in clockwise order; and zero if they are collinear. The */ /* result is also a rough approximation of twice the signed */ /* area of the triangle defined by the three points. */ /* */ /* Only the first and last routine should be used; the middle two are for */ /* timings. */ /* */ /* The last three use exact arithmetic to ensure a correct answer. The */ /* result returned is the determinant of a matrix. In orient2d() only, */ /* this determinant is computed adaptively, in the sense that exact */ /* arithmetic is used only to the degree it is needed to ensure that the */ /* returned value has the correct sign. Hence, orient2d() is usually quite */ /* fast, but will run more slowly when the input points are collinear or */ /* nearly so. */ /* */ /*****************************************************************************/ REAL orient2dadapt(const REAL *pa, const REAL *pb, const REAL *pc, const REAL detsum) { INEXACT REAL acx, acy, bcx, bcy; REAL acxtail, acytail, bcxtail, bcytail; INEXACT REAL detleft, detright; REAL detlefttail, detrighttail; REAL det, errbound; REAL B[4], C1[8], C2[12], D[16]; INEXACT REAL B3; int C1length, C2length, Dlength; REAL u[4]; INEXACT REAL u3; INEXACT REAL s1, t1; REAL s0, t0; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; acx = (REAL) (pa[0] - pc[0]); bcx = (REAL) (pb[0] - pc[0]); acy = (REAL) (pa[1] - pc[1]); bcy = (REAL) (pb[1] - pc[1]); Two_Product(acx, bcy, detleft, detlefttail); Two_Product(acy, bcx, detright, detrighttail); Two_Two_Diff(detleft, detlefttail, detright, detrighttail, B3, B[2], B[1], B[0]); B[3] = B3; det = estimate(4, B); errbound = ccwerrboundB * detsum; if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Diff_Tail(pa[0], pc[0], acx, acxtail); Two_Diff_Tail(pb[0], pc[0], bcx, bcxtail); Two_Diff_Tail(pa[1], pc[1], acy, acytail); Two_Diff_Tail(pb[1], pc[1], bcy, bcytail); if ((acxtail == 0.0) && (acytail == 0.0) && (bcxtail == 0.0) && (bcytail == 0.0)) { return det; } errbound = ccwerrboundC * detsum + resulterrbound * Absolute(det); det += (acx * bcytail + bcy * acxtail) - (acy * bcxtail + bcx * acytail); if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Product(acxtail, bcy, s1, s0); Two_Product(acytail, bcx, t1, t0); Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]); u[3] = u3; C1length = fast_expansion_sum_zeroelim(4, B, 4, u, C1); Two_Product(acx, bcytail, s1, s0); Two_Product(acy, bcxtail, t1, t0); Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]); u[3] = u3; C2length = fast_expansion_sum_zeroelim(C1length, C1, 4, u, C2); Two_Product(acxtail, bcytail, s1, s0); Two_Product(acytail, bcxtail, t1, t0); Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]); u[3] = u3; Dlength = fast_expansion_sum_zeroelim(C2length, C2, 4, u, D); return(D[Dlength - 1]); } REAL orient2d(const REAL *pa, const REAL *pb, const REAL *pc) { REAL detleft, detright, det; REAL detsum, errbound; REAL orient; FPU_ROUND_DOUBLE; detleft = (pa[0] - pc[0]) * (pb[1] - pc[1]); detright = (pa[1] - pc[1]) * (pb[0] - pc[0]); det = detleft - detright; if (detleft > 0.0) { if (detright <= 0.0) { FPU_RESTORE; return det; } else { detsum = detleft + detright; } } else if (detleft < 0.0) { if (detright >= 0.0) { FPU_RESTORE; return det; } else { detsum = -detleft - detright; } } else { FPU_RESTORE; return det; } errbound = ccwerrboundA * detsum; if ((det >= errbound) || (-det >= errbound)) { FPU_RESTORE; return det; } orient = orient2dadapt(pa, pb, pc, detsum); FPU_RESTORE; return orient; } REAL orient2d(const REAL ax, const REAL ay, const REAL bx, const REAL by, const REAL cx, const REAL cy) { REAL detleft, detright, det; REAL detsum, errbound; REAL orient; FPU_ROUND_DOUBLE; detleft = (ax - cx) * (by - cy); detright = (ay - cy) * (bx - cx); det = detleft - detright; if (detleft > 0.0) { if (detright <= 0.0) { FPU_RESTORE; return det; } else { detsum = detleft + detright; } } else if (detleft < 0.0) { if (detright >= 0.0) { FPU_RESTORE; return det; } else { detsum = -detleft - detright; } } else { FPU_RESTORE; return det; } errbound = ccwerrboundA * detsum; if ((det >= errbound) || (-det >= errbound)) { FPU_RESTORE; return det; } REAL pa[]={ax,ay}; REAL pb[]={bx,by}; REAL pc[]={cx,cy}; orient = orient2dadapt(pa, pb, pc, detsum); FPU_RESTORE; return orient; } /*****************************************************************************/ /* */ /* orient3dfast() Approximate 3D orientation test. Nonrobust. */ /* orient3dexact() Exact 3D orientation test. Robust. */ /* orient3dslow() Another exact 3D orientation test. Robust. */ /* orient3d() Adaptive exact 3D orientation test. Robust. */ /* */ /* Return a positive value if the point pd lies below the */ /* plane passing through pa, pb, and pc; "below" is defined so */ /* that pa, pb, and pc appear in counterclockwise order when */ /* viewed from above the plane. Returns a negative value if */ /* pd lies above the plane. Returns zero if the points are */ /* coplanar. The result is also a rough approximation of six */ /* times the signed volume of the tetrahedron defined by the */ /* four points. */ /* */ /* Only the first and last routine should be used; the middle two are for */ /* timings. */ /* */ /* The last three use exact arithmetic to ensure a correct answer. The */ /* result returned is the determinant of a matrix. In orient3d() only, */ /* this determinant is computed adaptively, in the sense that exact */ /* arithmetic is used only to the degree it is needed to ensure that the */ /* returned value has the correct sign. Hence, orient3d() is usually quite */ /* fast, but will run more slowly when the input points are coplanar or */ /* nearly so. */ /* */ /*****************************************************************************/ static REAL orient3dadapt(const REAL *pa, const REAL *pb, const REAL *pc, const REAL *pd, REAL permanent) { INEXACT REAL adx, bdx, cdx, ady, bdy, cdy, adz, bdz, cdz; REAL det, errbound; INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; REAL bc[4], ca[4], ab[4]; INEXACT REAL bc3, ca3, ab3; REAL adet[8], bdet[8], cdet[8]; int alen, blen, clen; REAL abdet[16]; int ablen; REAL *finnow, *finother, *finswap; REAL fin1[192], fin2[192]; int finlength; REAL adxtail, bdxtail, cdxtail; REAL adytail, bdytail, cdytail; REAL adztail, bdztail, cdztail; INEXACT REAL at_blarge, at_clarge; INEXACT REAL bt_clarge, bt_alarge; INEXACT REAL ct_alarge, ct_blarge; REAL at_b[4], at_c[4], bt_c[4], bt_a[4], ct_a[4], ct_b[4]; int at_blen, at_clen, bt_clen, bt_alen, ct_alen, ct_blen; INEXACT REAL bdxt_cdy1, cdxt_bdy1, cdxt_ady1; INEXACT REAL adxt_cdy1, adxt_bdy1, bdxt_ady1; REAL bdxt_cdy0, cdxt_bdy0, cdxt_ady0; REAL adxt_cdy0, adxt_bdy0, bdxt_ady0; INEXACT REAL bdyt_cdx1, cdyt_bdx1, cdyt_adx1; INEXACT REAL adyt_cdx1, adyt_bdx1, bdyt_adx1; REAL bdyt_cdx0, cdyt_bdx0, cdyt_adx0; REAL adyt_cdx0, adyt_bdx0, bdyt_adx0; REAL bct[8], cat[8], abt[8]; int bctlen, catlen, abtlen; INEXACT REAL bdxt_cdyt1, cdxt_bdyt1, cdxt_adyt1; INEXACT REAL adxt_cdyt1, adxt_bdyt1, bdxt_adyt1; REAL bdxt_cdyt0, cdxt_bdyt0, cdxt_adyt0; REAL adxt_cdyt0, adxt_bdyt0, bdxt_adyt0; REAL u[4], v[12], w[16]; INEXACT REAL u3; int vlength, wlength; REAL negate; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j, _k; REAL _0; adx = (REAL) (pa[0] - pd[0]); bdx = (REAL) (pb[0] - pd[0]); cdx = (REAL) (pc[0] - pd[0]); ady = (REAL) (pa[1] - pd[1]); bdy = (REAL) (pb[1] - pd[1]); cdy = (REAL) (pc[1] - pd[1]); adz = (REAL) (pa[2] - pd[2]); bdz = (REAL) (pb[2] - pd[2]); cdz = (REAL) (pc[2] - pd[2]); Two_Product(bdx, cdy, bdxcdy1, bdxcdy0); Two_Product(cdx, bdy, cdxbdy1, cdxbdy0); Two_Two_Diff(bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0]); bc[3] = bc3; alen = scale_expansion_zeroelim(4, bc, adz, adet); Two_Product(cdx, ady, cdxady1, cdxady0); Two_Product(adx, cdy, adxcdy1, adxcdy0); Two_Two_Diff(cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0]); ca[3] = ca3; blen = scale_expansion_zeroelim(4, ca, bdz, bdet); Two_Product(adx, bdy, adxbdy1, adxbdy0); Two_Product(bdx, ady, bdxady1, bdxady0); Two_Two_Diff(adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0]); ab[3] = ab3; clen = scale_expansion_zeroelim(4, ab, cdz, cdet); ablen = fast_expansion_sum_zeroelim(alen, adet, blen, bdet, abdet); finlength = fast_expansion_sum_zeroelim(ablen, abdet, clen, cdet, fin1); det = estimate(finlength, fin1); errbound = o3derrboundB * permanent; if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Diff_Tail(pa[0], pd[0], adx, adxtail); Two_Diff_Tail(pb[0], pd[0], bdx, bdxtail); Two_Diff_Tail(pc[0], pd[0], cdx, cdxtail); Two_Diff_Tail(pa[1], pd[1], ady, adytail); Two_Diff_Tail(pb[1], pd[1], bdy, bdytail); Two_Diff_Tail(pc[1], pd[1], cdy, cdytail); Two_Diff_Tail(pa[2], pd[2], adz, adztail); Two_Diff_Tail(pb[2], pd[2], bdz, bdztail); Two_Diff_Tail(pc[2], pd[2], cdz, cdztail); if ((adxtail == 0.0) && (bdxtail == 0.0) && (cdxtail == 0.0) && (adytail == 0.0) && (bdytail == 0.0) && (cdytail == 0.0) && (adztail == 0.0) && (bdztail == 0.0) && (cdztail == 0.0)) { return det; } errbound = o3derrboundC * permanent + resulterrbound * Absolute(det); det += (adz * ((bdx * cdytail + cdy * bdxtail) - (bdy * cdxtail + cdx * bdytail)) + adztail * (bdx * cdy - bdy * cdx)) + (bdz * ((cdx * adytail + ady * cdxtail) - (cdy * adxtail + adx * cdytail)) + bdztail * (cdx * ady - cdy * adx)) + (cdz * ((adx * bdytail + bdy * adxtail) - (ady * bdxtail + bdx * adytail)) + cdztail * (adx * bdy - ady * bdx)); if ((det >= errbound) || (-det >= errbound)) { return det; } finnow = fin1; finother = fin2; if (adxtail == 0.0) { if (adytail == 0.0) { at_b[0] = 0.0; at_blen = 1; at_c[0] = 0.0; at_clen = 1; } else { negate = -adytail; Two_Product(negate, bdx, at_blarge, at_b[0]); at_b[1] = at_blarge; at_blen = 2; Two_Product(adytail, cdx, at_clarge, at_c[0]); at_c[1] = at_clarge; at_clen = 2; } } else { if (adytail == 0.0) { Two_Product(adxtail, bdy, at_blarge, at_b[0]); at_b[1] = at_blarge; at_blen = 2; negate = -adxtail; Two_Product(negate, cdy, at_clarge, at_c[0]); at_c[1] = at_clarge; at_clen = 2; } else { Two_Product(adxtail, bdy, adxt_bdy1, adxt_bdy0); Two_Product(adytail, bdx, adyt_bdx1, adyt_bdx0); Two_Two_Diff(adxt_bdy1, adxt_bdy0, adyt_bdx1, adyt_bdx0, at_blarge, at_b[2], at_b[1], at_b[0]); at_b[3] = at_blarge; at_blen = 4; Two_Product(adytail, cdx, adyt_cdx1, adyt_cdx0); Two_Product(adxtail, cdy, adxt_cdy1, adxt_cdy0); Two_Two_Diff(adyt_cdx1, adyt_cdx0, adxt_cdy1, adxt_cdy0, at_clarge, at_c[2], at_c[1], at_c[0]); at_c[3] = at_clarge; at_clen = 4; } } if (bdxtail == 0.0) { if (bdytail == 0.0) { bt_c[0] = 0.0; bt_clen = 1; bt_a[0] = 0.0; bt_alen = 1; } else { negate = -bdytail; Two_Product(negate, cdx, bt_clarge, bt_c[0]); bt_c[1] = bt_clarge; bt_clen = 2; Two_Product(bdytail, adx, bt_alarge, bt_a[0]); bt_a[1] = bt_alarge; bt_alen = 2; } } else { if (bdytail == 0.0) { Two_Product(bdxtail, cdy, bt_clarge, bt_c[0]); bt_c[1] = bt_clarge; bt_clen = 2; negate = -bdxtail; Two_Product(negate, ady, bt_alarge, bt_a[0]); bt_a[1] = bt_alarge; bt_alen = 2; } else { Two_Product(bdxtail, cdy, bdxt_cdy1, bdxt_cdy0); Two_Product(bdytail, cdx, bdyt_cdx1, bdyt_cdx0); Two_Two_Diff(bdxt_cdy1, bdxt_cdy0, bdyt_cdx1, bdyt_cdx0, bt_clarge, bt_c[2], bt_c[1], bt_c[0]); bt_c[3] = bt_clarge; bt_clen = 4; Two_Product(bdytail, adx, bdyt_adx1, bdyt_adx0); Two_Product(bdxtail, ady, bdxt_ady1, bdxt_ady0); Two_Two_Diff(bdyt_adx1, bdyt_adx0, bdxt_ady1, bdxt_ady0, bt_alarge, bt_a[2], bt_a[1], bt_a[0]); bt_a[3] = bt_alarge; bt_alen = 4; } } if (cdxtail == 0.0) { if (cdytail == 0.0) { ct_a[0] = 0.0; ct_alen = 1; ct_b[0] = 0.0; ct_blen = 1; } else { negate = -cdytail; Two_Product(negate, adx, ct_alarge, ct_a[0]); ct_a[1] = ct_alarge; ct_alen = 2; Two_Product(cdytail, bdx, ct_blarge, ct_b[0]); ct_b[1] = ct_blarge; ct_blen = 2; } } else { if (cdytail == 0.0) { Two_Product(cdxtail, ady, ct_alarge, ct_a[0]); ct_a[1] = ct_alarge; ct_alen = 2; negate = -cdxtail; Two_Product(negate, bdy, ct_blarge, ct_b[0]); ct_b[1] = ct_blarge; ct_blen = 2; } else { Two_Product(cdxtail, ady, cdxt_ady1, cdxt_ady0); Two_Product(cdytail, adx, cdyt_adx1, cdyt_adx0); Two_Two_Diff(cdxt_ady1, cdxt_ady0, cdyt_adx1, cdyt_adx0, ct_alarge, ct_a[2], ct_a[1], ct_a[0]); ct_a[3] = ct_alarge; ct_alen = 4; Two_Product(cdytail, bdx, cdyt_bdx1, cdyt_bdx0); Two_Product(cdxtail, bdy, cdxt_bdy1, cdxt_bdy0); Two_Two_Diff(cdyt_bdx1, cdyt_bdx0, cdxt_bdy1, cdxt_bdy0, ct_blarge, ct_b[2], ct_b[1], ct_b[0]); ct_b[3] = ct_blarge; ct_blen = 4; } } bctlen = fast_expansion_sum_zeroelim(bt_clen, bt_c, ct_blen, ct_b, bct); wlength = scale_expansion_zeroelim(bctlen, bct, adz, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; catlen = fast_expansion_sum_zeroelim(ct_alen, ct_a, at_clen, at_c, cat); wlength = scale_expansion_zeroelim(catlen, cat, bdz, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; abtlen = fast_expansion_sum_zeroelim(at_blen, at_b, bt_alen, bt_a, abt); wlength = scale_expansion_zeroelim(abtlen, abt, cdz, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; if (adztail != 0.0) { vlength = scale_expansion_zeroelim(4, bc, adztail, v); finlength = fast_expansion_sum_zeroelim(finlength, finnow, vlength, v, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdztail != 0.0) { vlength = scale_expansion_zeroelim(4, ca, bdztail, v); finlength = fast_expansion_sum_zeroelim(finlength, finnow, vlength, v, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdztail != 0.0) { vlength = scale_expansion_zeroelim(4, ab, cdztail, v); finlength = fast_expansion_sum_zeroelim(finlength, finnow, vlength, v, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adxtail != 0.0) { if (bdytail != 0.0) { Two_Product(adxtail, bdytail, adxt_bdyt1, adxt_bdyt0); Two_One_Product(adxt_bdyt1, adxt_bdyt0, cdz, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (cdztail != 0.0) { Two_One_Product(adxt_bdyt1, adxt_bdyt0, cdztail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } if (cdytail != 0.0) { negate = -adxtail; Two_Product(negate, cdytail, adxt_cdyt1, adxt_cdyt0); Two_One_Product(adxt_cdyt1, adxt_cdyt0, bdz, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (bdztail != 0.0) { Two_One_Product(adxt_cdyt1, adxt_cdyt0, bdztail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } } if (bdxtail != 0.0) { if (cdytail != 0.0) { Two_Product(bdxtail, cdytail, bdxt_cdyt1, bdxt_cdyt0); Two_One_Product(bdxt_cdyt1, bdxt_cdyt0, adz, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (adztail != 0.0) { Two_One_Product(bdxt_cdyt1, bdxt_cdyt0, adztail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } if (adytail != 0.0) { negate = -bdxtail; Two_Product(negate, adytail, bdxt_adyt1, bdxt_adyt0); Two_One_Product(bdxt_adyt1, bdxt_adyt0, cdz, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (cdztail != 0.0) { Two_One_Product(bdxt_adyt1, bdxt_adyt0, cdztail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } } if (cdxtail != 0.0) { if (adytail != 0.0) { Two_Product(cdxtail, adytail, cdxt_adyt1, cdxt_adyt0); Two_One_Product(cdxt_adyt1, cdxt_adyt0, bdz, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (bdztail != 0.0) { Two_One_Product(cdxt_adyt1, cdxt_adyt0, bdztail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } if (bdytail != 0.0) { negate = -cdxtail; Two_Product(negate, bdytail, cdxt_bdyt1, cdxt_bdyt0); Two_One_Product(cdxt_bdyt1, cdxt_bdyt0, adz, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (adztail != 0.0) { Two_One_Product(cdxt_bdyt1, cdxt_bdyt0, adztail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } } if (adztail != 0.0) { wlength = scale_expansion_zeroelim(bctlen, bct, adztail, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdztail != 0.0) { wlength = scale_expansion_zeroelim(catlen, cat, bdztail, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdztail != 0.0) { wlength = scale_expansion_zeroelim(abtlen, abt, cdztail, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; } return finnow[finlength - 1]; } REAL orient3d(const REAL *pa, const REAL *pb, const REAL *pc, const REAL *pd) { REAL adx, bdx, cdx, ady, bdy, cdy, adz, bdz, cdz; REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; REAL det; REAL permanent, errbound; REAL orient; FPU_ROUND_DOUBLE; adx = pa[0] - pd[0]; bdx = pb[0] - pd[0]; cdx = pc[0] - pd[0]; ady = pa[1] - pd[1]; bdy = pb[1] - pd[1]; cdy = pc[1] - pd[1]; adz = pa[2] - pd[2]; bdz = pb[2] - pd[2]; cdz = pc[2] - pd[2]; bdxcdy = bdx * cdy; cdxbdy = cdx * bdy; cdxady = cdx * ady; adxcdy = adx * cdy; adxbdy = adx * bdy; bdxady = bdx * ady; det = adz * (bdxcdy - cdxbdy) + bdz * (cdxady - adxcdy) + cdz * (adxbdy - bdxady); permanent = (Absolute(bdxcdy) + Absolute(cdxbdy)) * Absolute(adz) + (Absolute(cdxady) + Absolute(adxcdy)) * Absolute(bdz) + (Absolute(adxbdy) + Absolute(bdxady)) * Absolute(cdz); errbound = o3derrboundA * permanent; if ((det > errbound) || (-det > errbound)) { FPU_RESTORE; return det; } orient = orient3dadapt(pa, pb, pc, pd, permanent); FPU_RESTORE; return orient; } /*****************************************************************************/ /* */ /* incirclefast() Approximate 2D incircle test. Nonrobust. */ /* incircleexact() Exact 2D incircle test. Robust. */ /* incircleslow() Another exact 2D incircle test. Robust. */ /* incircle() Adaptive exact 2D incircle test. Robust. */ /* */ /* Return a positive value if the point pd lies inside the */ /* circle passing through pa, pb, and pc; a negative value if */ /* it lies outside; and zero if the four points are cocircular.*/ /* The points pa, pb, and pc must be in counterclockwise */ /* order, or the sign of the result will be reversed. */ /* */ /* Only the first and last routine should be used; the middle two are for */ /* timings. */ /* */ /* The last three use exact arithmetic to ensure a correct answer. The */ /* result returned is the determinant of a matrix. In incircle() only, */ /* this determinant is computed adaptively, in the sense that exact */ /* arithmetic is used only to the degree it is needed to ensure that the */ /* returned value has the correct sign. Hence, incircle() is usually quite */ /* fast, but will run more slowly when the input points are cocircular or */ /* nearly so. */ /* */ /*****************************************************************************/ static REAL incircleadapt(const REAL *pa, const REAL *pb, const REAL *pc, const REAL *pd, REAL permanent) { INEXACT REAL adx, bdx, cdx, ady, bdy, cdy; REAL det, errbound; INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; REAL bc[4], ca[4], ab[4]; INEXACT REAL bc3, ca3, ab3; REAL axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32]; int axbclen, axxbclen, aybclen, ayybclen, alen; REAL bxca[8], bxxca[16], byca[8], byyca[16], bdet[32]; int bxcalen, bxxcalen, bycalen, byycalen, blen; REAL cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32]; int cxablen, cxxablen, cyablen, cyyablen, clen; REAL abdet[64]; int ablen; REAL fin1[1152], fin2[1152]; REAL *finnow, *finother, *finswap; int finlength; REAL adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail; INEXACT REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1; REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0; REAL aa[4], bb[4], cc[4]; INEXACT REAL aa3, bb3, cc3; INEXACT REAL ti1, tj1; REAL ti0, tj0; REAL u[4], v[4]; INEXACT REAL u3, v3; REAL temp8[8], temp16a[16], temp16b[16], temp16c[16]; REAL temp32a[32], temp32b[32], temp48[48], temp64[64]; int temp8len, temp16alen, temp16blen, temp16clen; int temp32alen, temp32blen, temp48len, temp64len; REAL axtbb[8], axtcc[8], aytbb[8], aytcc[8]; int axtbblen, axtcclen, aytbblen, aytcclen; REAL bxtaa[8], bxtcc[8], bytaa[8], bytcc[8]; int bxtaalen, bxtcclen, bytaalen, bytcclen; REAL cxtaa[8], cxtbb[8], cytaa[8], cytbb[8]; int cxtaalen, cxtbblen, cytaalen, cytbblen; REAL axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8]; int axtbclen = 0, aytbclen = 0; int bxtcalen = 0, bytcalen = 0; int cxtablen = 0, cytablen = 0; REAL axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16]; int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen; REAL axtbctt[8], aytbctt[8], bxtcatt[8]; REAL bytcatt[8], cxtabtt[8], cytabtt[8]; int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen; REAL abt[8], bct[8], cat[8]; int abtlen, bctlen, catlen; REAL abtt[4], bctt[4], catt[4]; int abttlen, bcttlen, cattlen; INEXACT REAL abtt3, bctt3, catt3; REAL negate; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; adx = (REAL) (pa[0] - pd[0]); bdx = (REAL) (pb[0] - pd[0]); cdx = (REAL) (pc[0] - pd[0]); ady = (REAL) (pa[1] - pd[1]); bdy = (REAL) (pb[1] - pd[1]); cdy = (REAL) (pc[1] - pd[1]); Two_Product(bdx, cdy, bdxcdy1, bdxcdy0); Two_Product(cdx, bdy, cdxbdy1, cdxbdy0); Two_Two_Diff(bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0]); bc[3] = bc3; axbclen = scale_expansion_zeroelim(4, bc, adx, axbc); axxbclen = scale_expansion_zeroelim(axbclen, axbc, adx, axxbc); aybclen = scale_expansion_zeroelim(4, bc, ady, aybc); ayybclen = scale_expansion_zeroelim(aybclen, aybc, ady, ayybc); alen = fast_expansion_sum_zeroelim(axxbclen, axxbc, ayybclen, ayybc, adet); Two_Product(cdx, ady, cdxady1, cdxady0); Two_Product(adx, cdy, adxcdy1, adxcdy0); Two_Two_Diff(cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0]); ca[3] = ca3; bxcalen = scale_expansion_zeroelim(4, ca, bdx, bxca); bxxcalen = scale_expansion_zeroelim(bxcalen, bxca, bdx, bxxca); bycalen = scale_expansion_zeroelim(4, ca, bdy, byca); byycalen = scale_expansion_zeroelim(bycalen, byca, bdy, byyca); blen = fast_expansion_sum_zeroelim(bxxcalen, bxxca, byycalen, byyca, bdet); Two_Product(adx, bdy, adxbdy1, adxbdy0); Two_Product(bdx, ady, bdxady1, bdxady0); Two_Two_Diff(adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0]); ab[3] = ab3; cxablen = scale_expansion_zeroelim(4, ab, cdx, cxab); cxxablen = scale_expansion_zeroelim(cxablen, cxab, cdx, cxxab); cyablen = scale_expansion_zeroelim(4, ab, cdy, cyab); cyyablen = scale_expansion_zeroelim(cyablen, cyab, cdy, cyyab); clen = fast_expansion_sum_zeroelim(cxxablen, cxxab, cyyablen, cyyab, cdet); ablen = fast_expansion_sum_zeroelim(alen, adet, blen, bdet, abdet); finlength = fast_expansion_sum_zeroelim(ablen, abdet, clen, cdet, fin1); det = estimate(finlength, fin1); errbound = iccerrboundB * permanent; if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Diff_Tail(pa[0], pd[0], adx, adxtail); Two_Diff_Tail(pa[1], pd[1], ady, adytail); Two_Diff_Tail(pb[0], pd[0], bdx, bdxtail); Two_Diff_Tail(pb[1], pd[1], bdy, bdytail); Two_Diff_Tail(pc[0], pd[0], cdx, cdxtail); Two_Diff_Tail(pc[1], pd[1], cdy, cdytail); if ((adxtail == 0.0) && (bdxtail == 0.0) && (cdxtail == 0.0) && (adytail == 0.0) && (bdytail == 0.0) && (cdytail == 0.0)) { return det; } errbound = iccerrboundC * permanent + resulterrbound * Absolute(det); det += ((adx * adx + ady * ady) * ((bdx * cdytail + cdy * bdxtail) - (bdy * cdxtail + cdx * bdytail)) + 2.0 * (adx * adxtail + ady * adytail) * (bdx * cdy - bdy * cdx)) + ((bdx * bdx + bdy * bdy) * ((cdx * adytail + ady * cdxtail) - (cdy * adxtail + adx * cdytail)) + 2.0 * (bdx * bdxtail + bdy * bdytail) * (cdx * ady - cdy * adx)) + ((cdx * cdx + cdy * cdy) * ((adx * bdytail + bdy * adxtail) - (ady * bdxtail + bdx * adytail)) + 2.0 * (cdx * cdxtail + cdy * cdytail) * (adx * bdy - ady * bdx)); if ((det >= errbound) || (-det >= errbound)) { return det; } finnow = fin1; finother = fin2; if ((bdxtail != 0.0) || (bdytail != 0.0) || (cdxtail != 0.0) || (cdytail != 0.0)) { Square(adx, adxadx1, adxadx0); Square(ady, adyady1, adyady0); Two_Two_Sum(adxadx1, adxadx0, adyady1, adyady0, aa3, aa[2], aa[1], aa[0]); aa[3] = aa3; } if ((cdxtail != 0.0) || (cdytail != 0.0) || (adxtail != 0.0) || (adytail != 0.0)) { Square(bdx, bdxbdx1, bdxbdx0); Square(bdy, bdybdy1, bdybdy0); Two_Two_Sum(bdxbdx1, bdxbdx0, bdybdy1, bdybdy0, bb3, bb[2], bb[1], bb[0]); bb[3] = bb3; } if ((adxtail != 0.0) || (adytail != 0.0) || (bdxtail != 0.0) || (bdytail != 0.0)) { Square(cdx, cdxcdx1, cdxcdx0); Square(cdy, cdycdy1, cdycdy0); Two_Two_Sum(cdxcdx1, cdxcdx0, cdycdy1, cdycdy0, cc3, cc[2], cc[1], cc[0]); cc[3] = cc3; } if (adxtail != 0.0) { axtbclen = scale_expansion_zeroelim(4, bc, adxtail, axtbc); temp16alen = scale_expansion_zeroelim(axtbclen, axtbc, 2.0 * adx, temp16a); axtcclen = scale_expansion_zeroelim(4, cc, adxtail, axtcc); temp16blen = scale_expansion_zeroelim(axtcclen, axtcc, bdy, temp16b); axtbblen = scale_expansion_zeroelim(4, bb, adxtail, axtbb); temp16clen = scale_expansion_zeroelim(axtbblen, axtbb, -cdy, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adytail != 0.0) { aytbclen = scale_expansion_zeroelim(4, bc, adytail, aytbc); temp16alen = scale_expansion_zeroelim(aytbclen, aytbc, 2.0 * ady, temp16a); aytbblen = scale_expansion_zeroelim(4, bb, adytail, aytbb); temp16blen = scale_expansion_zeroelim(aytbblen, aytbb, cdx, temp16b); aytcclen = scale_expansion_zeroelim(4, cc, adytail, aytcc); temp16clen = scale_expansion_zeroelim(aytcclen, aytcc, -bdx, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdxtail != 0.0) { bxtcalen = scale_expansion_zeroelim(4, ca, bdxtail, bxtca); temp16alen = scale_expansion_zeroelim(bxtcalen, bxtca, 2.0 * bdx, temp16a); bxtaalen = scale_expansion_zeroelim(4, aa, bdxtail, bxtaa); temp16blen = scale_expansion_zeroelim(bxtaalen, bxtaa, cdy, temp16b); bxtcclen = scale_expansion_zeroelim(4, cc, bdxtail, bxtcc); temp16clen = scale_expansion_zeroelim(bxtcclen, bxtcc, -ady, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdytail != 0.0) { bytcalen = scale_expansion_zeroelim(4, ca, bdytail, bytca); temp16alen = scale_expansion_zeroelim(bytcalen, bytca, 2.0 * bdy, temp16a); bytcclen = scale_expansion_zeroelim(4, cc, bdytail, bytcc); temp16blen = scale_expansion_zeroelim(bytcclen, bytcc, adx, temp16b); bytaalen = scale_expansion_zeroelim(4, aa, bdytail, bytaa); temp16clen = scale_expansion_zeroelim(bytaalen, bytaa, -cdx, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdxtail != 0.0) { cxtablen = scale_expansion_zeroelim(4, ab, cdxtail, cxtab); temp16alen = scale_expansion_zeroelim(cxtablen, cxtab, 2.0 * cdx, temp16a); cxtbblen = scale_expansion_zeroelim(4, bb, cdxtail, cxtbb); temp16blen = scale_expansion_zeroelim(cxtbblen, cxtbb, ady, temp16b); cxtaalen = scale_expansion_zeroelim(4, aa, cdxtail, cxtaa); temp16clen = scale_expansion_zeroelim(cxtaalen, cxtaa, -bdy, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdytail != 0.0) { cytablen = scale_expansion_zeroelim(4, ab, cdytail, cytab); temp16alen = scale_expansion_zeroelim(cytablen, cytab, 2.0 * cdy, temp16a); cytaalen = scale_expansion_zeroelim(4, aa, cdytail, cytaa); temp16blen = scale_expansion_zeroelim(cytaalen, cytaa, bdx, temp16b); cytbblen = scale_expansion_zeroelim(4, bb, cdytail, cytbb); temp16clen = scale_expansion_zeroelim(cytbblen, cytbb, -adx, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if ((adxtail != 0.0) || (adytail != 0.0)) { if ((bdxtail != 0.0) || (bdytail != 0.0) || (cdxtail != 0.0) || (cdytail != 0.0)) { Two_Product(bdxtail, cdy, ti1, ti0); Two_Product(bdx, cdytail, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]); u[3] = u3; negate = -bdy; Two_Product(cdxtail, negate, ti1, ti0); negate = -bdytail; Two_Product(cdx, negate, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]); v[3] = v3; bctlen = fast_expansion_sum_zeroelim(4, u, 4, v, bct); Two_Product(bdxtail, cdytail, ti1, ti0); Two_Product(cdxtail, bdytail, tj1, tj0); Two_Two_Diff(ti1, ti0, tj1, tj0, bctt3, bctt[2], bctt[1], bctt[0]); bctt[3] = bctt3; bcttlen = 4; } else { bct[0] = 0.0; bctlen = 1; bctt[0] = 0.0; bcttlen = 1; } if (adxtail != 0.0) { temp16alen = scale_expansion_zeroelim(axtbclen, axtbc, adxtail, temp16a); axtbctlen = scale_expansion_zeroelim(bctlen, bct, adxtail, axtbct); temp32alen = scale_expansion_zeroelim(axtbctlen, axtbct, 2.0 * adx, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; if (bdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, cc, adxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, bdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, bb, -adxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, cdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim(axtbctlen, axtbct, adxtail, temp32a); axtbcttlen = scale_expansion_zeroelim(bcttlen, bctt, adxtail, axtbctt); temp16alen = scale_expansion_zeroelim(axtbcttlen, axtbctt, 2.0 * adx, temp16a); temp16blen = scale_expansion_zeroelim(axtbcttlen, axtbctt, adxtail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adytail != 0.0) { temp16alen = scale_expansion_zeroelim(aytbclen, aytbc, adytail, temp16a); aytbctlen = scale_expansion_zeroelim(bctlen, bct, adytail, aytbct); temp32alen = scale_expansion_zeroelim(aytbctlen, aytbct, 2.0 * ady, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim(aytbctlen, aytbct, adytail, temp32a); aytbcttlen = scale_expansion_zeroelim(bcttlen, bctt, adytail, aytbctt); temp16alen = scale_expansion_zeroelim(aytbcttlen, aytbctt, 2.0 * ady, temp16a); temp16blen = scale_expansion_zeroelim(aytbcttlen, aytbctt, adytail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } } if ((bdxtail != 0.0) || (bdytail != 0.0)) { if ((cdxtail != 0.0) || (cdytail != 0.0) || (adxtail != 0.0) || (adytail != 0.0)) { Two_Product(cdxtail, ady, ti1, ti0); Two_Product(cdx, adytail, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]); u[3] = u3; negate = -cdy; Two_Product(adxtail, negate, ti1, ti0); negate = -cdytail; Two_Product(adx, negate, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]); v[3] = v3; catlen = fast_expansion_sum_zeroelim(4, u, 4, v, cat); Two_Product(cdxtail, adytail, ti1, ti0); Two_Product(adxtail, cdytail, tj1, tj0); Two_Two_Diff(ti1, ti0, tj1, tj0, catt3, catt[2], catt[1], catt[0]); catt[3] = catt3; cattlen = 4; } else { cat[0] = 0.0; catlen = 1; catt[0] = 0.0; cattlen = 1; } if (bdxtail != 0.0) { temp16alen = scale_expansion_zeroelim(bxtcalen, bxtca, bdxtail, temp16a); bxtcatlen = scale_expansion_zeroelim(catlen, cat, bdxtail, bxtcat); temp32alen = scale_expansion_zeroelim(bxtcatlen, bxtcat, 2.0 * bdx, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; if (cdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, aa, bdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, cdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adytail != 0.0) { temp8len = scale_expansion_zeroelim(4, cc, -bdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, adytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim(bxtcatlen, bxtcat, bdxtail, temp32a); bxtcattlen = scale_expansion_zeroelim(cattlen, catt, bdxtail, bxtcatt); temp16alen = scale_expansion_zeroelim(bxtcattlen, bxtcatt, 2.0 * bdx, temp16a); temp16blen = scale_expansion_zeroelim(bxtcattlen, bxtcatt, bdxtail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdytail != 0.0) { temp16alen = scale_expansion_zeroelim(bytcalen, bytca, bdytail, temp16a); bytcatlen = scale_expansion_zeroelim(catlen, cat, bdytail, bytcat); temp32alen = scale_expansion_zeroelim(bytcatlen, bytcat, 2.0 * bdy, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim(bytcatlen, bytcat, bdytail, temp32a); bytcattlen = scale_expansion_zeroelim(cattlen, catt, bdytail, bytcatt); temp16alen = scale_expansion_zeroelim(bytcattlen, bytcatt, 2.0 * bdy, temp16a); temp16blen = scale_expansion_zeroelim(bytcattlen, bytcatt, bdytail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } } if ((cdxtail != 0.0) || (cdytail != 0.0)) { if ((adxtail != 0.0) || (adytail != 0.0) || (bdxtail != 0.0) || (bdytail != 0.0)) { Two_Product(adxtail, bdy, ti1, ti0); Two_Product(adx, bdytail, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]); u[3] = u3; negate = -ady; Two_Product(bdxtail, negate, ti1, ti0); negate = -adytail; Two_Product(bdx, negate, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]); v[3] = v3; abtlen = fast_expansion_sum_zeroelim(4, u, 4, v, abt); Two_Product(adxtail, bdytail, ti1, ti0); Two_Product(bdxtail, adytail, tj1, tj0); Two_Two_Diff(ti1, ti0, tj1, tj0, abtt3, abtt[2], abtt[1], abtt[0]); abtt[3] = abtt3; abttlen = 4; } else { abt[0] = 0.0; abtlen = 1; abtt[0] = 0.0; abttlen = 1; } if (cdxtail != 0.0) { temp16alen = scale_expansion_zeroelim(cxtablen, cxtab, cdxtail, temp16a); cxtabtlen = scale_expansion_zeroelim(abtlen, abt, cdxtail, cxtabt); temp32alen = scale_expansion_zeroelim(cxtabtlen, cxtabt, 2.0 * cdx, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; if (adytail != 0.0) { temp8len = scale_expansion_zeroelim(4, bb, cdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, adytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, aa, -cdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, bdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim(cxtabtlen, cxtabt, cdxtail, temp32a); cxtabttlen = scale_expansion_zeroelim(abttlen, abtt, cdxtail, cxtabtt); temp16alen = scale_expansion_zeroelim(cxtabttlen, cxtabtt, 2.0 * cdx, temp16a); temp16blen = scale_expansion_zeroelim(cxtabttlen, cxtabtt, cdxtail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdytail != 0.0) { temp16alen = scale_expansion_zeroelim(cytablen, cytab, cdytail, temp16a); cytabtlen = scale_expansion_zeroelim(abtlen, abt, cdytail, cytabt); temp32alen = scale_expansion_zeroelim(cytabtlen, cytabt, 2.0 * cdy, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim(cytabtlen, cytabt, cdytail, temp32a); cytabttlen = scale_expansion_zeroelim(abttlen, abtt, cdytail, cytabtt); temp16alen = scale_expansion_zeroelim(cytabttlen, cytabtt, 2.0 * cdy, temp16a); temp16blen = scale_expansion_zeroelim(cytabttlen, cytabtt, cdytail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } } return finnow[finlength - 1]; } REAL incircle(const REAL *pa, const REAL *pb, const REAL *pc, const REAL *pd) { REAL adx, bdx, cdx, ady, bdy, cdy; REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; REAL alift, blift, clift; REAL det; REAL permanent, errbound; REAL inc; FPU_ROUND_DOUBLE; adx = pa[0] - pd[0]; bdx = pb[0] - pd[0]; cdx = pc[0] - pd[0]; ady = pa[1] - pd[1]; bdy = pb[1] - pd[1]; cdy = pc[1] - pd[1]; bdxcdy = bdx * cdy; cdxbdy = cdx * bdy; alift = adx * adx + ady * ady; cdxady = cdx * ady; adxcdy = adx * cdy; blift = bdx * bdx + bdy * bdy; adxbdy = adx * bdy; bdxady = bdx * ady; clift = cdx * cdx + cdy * cdy; det = alift * (bdxcdy - cdxbdy) + blift * (cdxady - adxcdy) + clift * (adxbdy - bdxady); permanent = (Absolute(bdxcdy) + Absolute(cdxbdy)) * alift + (Absolute(cdxady) + Absolute(adxcdy)) * blift + (Absolute(adxbdy) + Absolute(bdxady)) * clift; errbound = iccerrboundA * permanent; if ((det > errbound) || (-det > errbound)) { FPU_RESTORE; return det; } inc = incircleadapt(pa, pb, pc, pd, permanent); FPU_RESTORE; return inc; } REAL incircle(REAL ax, REAL ay, REAL bx, REAL by, REAL cx, REAL cy, REAL dx, REAL dy) { REAL adx, bdx, cdx, ady, bdy, cdy; REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; REAL alift, blift, clift; REAL det; REAL permanent, errbound; REAL inc; FPU_ROUND_DOUBLE; adx = ax - dx; bdx = bx - dx; cdx = cx - dx; ady = ay - dy; bdy = by - dy; cdy = cy - dy; bdxcdy = bdx * cdy; cdxbdy = cdx * bdy; alift = adx * adx + ady * ady; cdxady = cdx * ady; adxcdy = adx * cdy; blift = bdx * bdx + bdy * bdy; adxbdy = adx * bdy; bdxady = bdx * ady; clift = cdx * cdx + cdy * cdy; det = alift * (bdxcdy - cdxbdy) + blift * (cdxady - adxcdy) + clift * (adxbdy - bdxady); permanent = (Absolute(bdxcdy) + Absolute(cdxbdy)) * alift + (Absolute(cdxady) + Absolute(adxcdy)) * blift + (Absolute(adxbdy) + Absolute(bdxady)) * clift; errbound = iccerrboundA * permanent; if ((det > errbound) || (-det > errbound)) { FPU_RESTORE; return det; } REAL pa[]={ax,ay}; REAL pb[]={bx,by}; REAL pc[]={cx,cy}; REAL pd[]={dx,dy}; inc = incircleadapt(pa, pb, pc, pd, permanent); FPU_RESTORE; return inc; } /*****************************************************************************/ /* */ /* inspherefast() Approximate 3D insphere test. Nonrobust. */ /* insphereexact() Exact 3D insphere test. Robust. */ /* insphereslow() Another exact 3D insphere test. Robust. */ /* insphere() Adaptive exact 3D insphere test. Robust. */ /* */ /* Return a positive value if the point pe lies inside the */ /* sphere passing through pa, pb, pc, and pd; a negative value */ /* if it lies outside; and zero if the five points are */ /* cospherical. The points pa, pb, pc, and pd must be ordered */ /* so that they have a positive orientation (as defined by */ /* orient3d()), or the sign of the result will be reversed. */ /* */ /* Only the first and last routine should be used; the middle two are for */ /* timings. */ /* */ /* The last three use exact arithmetic to ensure a correct answer. The */ /* result returned is the determinant of a matrix. In insphere() only, */ /* this determinant is computed adaptively, in the sense that exact */ /* arithmetic is used only to the degree it is needed to ensure that the */ /* returned value has the correct sign. Hence, insphere() is usually quite */ /* fast, but will run more slowly when the input points are cospherical or */ /* nearly so. */ /* */ /*****************************************************************************/ static REAL insphereexact(const REAL *pa, const REAL *pb, const REAL *pc, const REAL *pd, const REAL *pe) { INEXACT REAL axby1, bxcy1, cxdy1, dxey1, exay1; INEXACT REAL bxay1, cxby1, dxcy1, exdy1, axey1; INEXACT REAL axcy1, bxdy1, cxey1, dxay1, exby1; INEXACT REAL cxay1, dxby1, excy1, axdy1, bxey1; REAL axby0, bxcy0, cxdy0, dxey0, exay0; REAL bxay0, cxby0, dxcy0, exdy0, axey0; REAL axcy0, bxdy0, cxey0, dxay0, exby0; REAL cxay0, dxby0, excy0, axdy0, bxey0; REAL ab[4], bc[4], cd[4], de[4], ea[4]; REAL ac[4], bd[4], ce[4], da[4], eb[4]; REAL temp8a[8], temp8b[8], temp16[16]; int temp8alen, temp8blen, temp16len; REAL abc[24], bcd[24], cde[24], dea[24], eab[24]; REAL abd[24], bce[24], cda[24], deb[24], eac[24]; int abclen, bcdlen, cdelen, dealen, eablen; int abdlen, bcelen, cdalen, deblen, eaclen; REAL temp48a[48], temp48b[48]; int temp48alen, temp48blen; REAL abcd[96], bcde[96], cdea[96], deab[96], eabc[96]; int abcdlen, bcdelen, cdealen, deablen, eabclen; REAL temp192[192]; REAL det384x[384], det384y[384], det384z[384]; int xlen, ylen, zlen; REAL detxy[768]; int xylen; REAL adet[1152], bdet[1152], cdet[1152], ddet[1152], edet[1152]; int alen, blen, clen, dlen, elen; REAL abdet[2304], cddet[2304], cdedet[3456]; int ablen, cdlen; REAL deter[5760]; int deterlen; int i; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; Two_Product(pa[0], pb[1], axby1, axby0); Two_Product(pb[0], pa[1], bxay1, bxay0); Two_Two_Diff(axby1, axby0, bxay1, bxay0, ab[3], ab[2], ab[1], ab[0]); Two_Product(pb[0], pc[1], bxcy1, bxcy0); Two_Product(pc[0], pb[1], cxby1, cxby0); Two_Two_Diff(bxcy1, bxcy0, cxby1, cxby0, bc[3], bc[2], bc[1], bc[0]); Two_Product(pc[0], pd[1], cxdy1, cxdy0); Two_Product(pd[0], pc[1], dxcy1, dxcy0); Two_Two_Diff(cxdy1, cxdy0, dxcy1, dxcy0, cd[3], cd[2], cd[1], cd[0]); Two_Product(pd[0], pe[1], dxey1, dxey0); Two_Product(pe[0], pd[1], exdy1, exdy0); Two_Two_Diff(dxey1, dxey0, exdy1, exdy0, de[3], de[2], de[1], de[0]); Two_Product(pe[0], pa[1], exay1, exay0); Two_Product(pa[0], pe[1], axey1, axey0); Two_Two_Diff(exay1, exay0, axey1, axey0, ea[3], ea[2], ea[1], ea[0]); Two_Product(pa[0], pc[1], axcy1, axcy0); Two_Product(pc[0], pa[1], cxay1, cxay0); Two_Two_Diff(axcy1, axcy0, cxay1, cxay0, ac[3], ac[2], ac[1], ac[0]); Two_Product(pb[0], pd[1], bxdy1, bxdy0); Two_Product(pd[0], pb[1], dxby1, dxby0); Two_Two_Diff(bxdy1, bxdy0, dxby1, dxby0, bd[3], bd[2], bd[1], bd[0]); Two_Product(pc[0], pe[1], cxey1, cxey0); Two_Product(pe[0], pc[1], excy1, excy0); Two_Two_Diff(cxey1, cxey0, excy1, excy0, ce[3], ce[2], ce[1], ce[0]); Two_Product(pd[0], pa[1], dxay1, dxay0); Two_Product(pa[0], pd[1], axdy1, axdy0); Two_Two_Diff(dxay1, dxay0, axdy1, axdy0, da[3], da[2], da[1], da[0]); Two_Product(pe[0], pb[1], exby1, exby0); Two_Product(pb[0], pe[1], bxey1, bxey0); Two_Two_Diff(exby1, exby0, bxey1, bxey0, eb[3], eb[2], eb[1], eb[0]); temp8alen = scale_expansion_zeroelim(4, bc, pa[2], temp8a); temp8blen = scale_expansion_zeroelim(4, ac, -pb[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, ab, pc[2], temp8a); abclen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, abc); temp8alen = scale_expansion_zeroelim(4, cd, pb[2], temp8a); temp8blen = scale_expansion_zeroelim(4, bd, -pc[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, bc, pd[2], temp8a); bcdlen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, bcd); temp8alen = scale_expansion_zeroelim(4, de, pc[2], temp8a); temp8blen = scale_expansion_zeroelim(4, ce, -pd[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, cd, pe[2], temp8a); cdelen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, cde); temp8alen = scale_expansion_zeroelim(4, ea, pd[2], temp8a); temp8blen = scale_expansion_zeroelim(4, da, -pe[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, de, pa[2], temp8a); dealen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, dea); temp8alen = scale_expansion_zeroelim(4, ab, pe[2], temp8a); temp8blen = scale_expansion_zeroelim(4, eb, -pa[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, ea, pb[2], temp8a); eablen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, eab); temp8alen = scale_expansion_zeroelim(4, bd, pa[2], temp8a); temp8blen = scale_expansion_zeroelim(4, da, pb[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, ab, pd[2], temp8a); abdlen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, abd); temp8alen = scale_expansion_zeroelim(4, ce, pb[2], temp8a); temp8blen = scale_expansion_zeroelim(4, eb, pc[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, bc, pe[2], temp8a); bcelen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, bce); temp8alen = scale_expansion_zeroelim(4, da, pc[2], temp8a); temp8blen = scale_expansion_zeroelim(4, ac, pd[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, cd, pa[2], temp8a); cdalen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, cda); temp8alen = scale_expansion_zeroelim(4, eb, pd[2], temp8a); temp8blen = scale_expansion_zeroelim(4, bd, pe[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, de, pb[2], temp8a); deblen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, deb); temp8alen = scale_expansion_zeroelim(4, ac, pe[2], temp8a); temp8blen = scale_expansion_zeroelim(4, ce, pa[2], temp8b); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp8alen = scale_expansion_zeroelim(4, ea, pc[2], temp8a); eaclen = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp16len, temp16, eac); temp48alen = fast_expansion_sum_zeroelim(cdelen, cde, bcelen, bce, temp48a); temp48blen = fast_expansion_sum_zeroelim(deblen, deb, bcdlen, bcd, temp48b); for (i = 0; i < temp48blen; i++) { temp48b[i] = -temp48b[i]; } bcdelen = fast_expansion_sum_zeroelim(temp48alen, temp48a, temp48blen, temp48b, bcde); xlen = scale_expansion_zeroelim(bcdelen, bcde, pa[0], temp192); xlen = scale_expansion_zeroelim(xlen, temp192, pa[0], det384x); ylen = scale_expansion_zeroelim(bcdelen, bcde, pa[1], temp192); ylen = scale_expansion_zeroelim(ylen, temp192, pa[1], det384y); zlen = scale_expansion_zeroelim(bcdelen, bcde, pa[2], temp192); zlen = scale_expansion_zeroelim(zlen, temp192, pa[2], det384z); xylen = fast_expansion_sum_zeroelim(xlen, det384x, ylen, det384y, detxy); alen = fast_expansion_sum_zeroelim(xylen, detxy, zlen, det384z, adet); temp48alen = fast_expansion_sum_zeroelim(dealen, dea, cdalen, cda, temp48a); temp48blen = fast_expansion_sum_zeroelim(eaclen, eac, cdelen, cde, temp48b); for (i = 0; i < temp48blen; i++) { temp48b[i] = -temp48b[i]; } cdealen = fast_expansion_sum_zeroelim(temp48alen, temp48a, temp48blen, temp48b, cdea); xlen = scale_expansion_zeroelim(cdealen, cdea, pb[0], temp192); xlen = scale_expansion_zeroelim(xlen, temp192, pb[0], det384x); ylen = scale_expansion_zeroelim(cdealen, cdea, pb[1], temp192); ylen = scale_expansion_zeroelim(ylen, temp192, pb[1], det384y); zlen = scale_expansion_zeroelim(cdealen, cdea, pb[2], temp192); zlen = scale_expansion_zeroelim(zlen, temp192, pb[2], det384z); xylen = fast_expansion_sum_zeroelim(xlen, det384x, ylen, det384y, detxy); blen = fast_expansion_sum_zeroelim(xylen, detxy, zlen, det384z, bdet); temp48alen = fast_expansion_sum_zeroelim(eablen, eab, deblen, deb, temp48a); temp48blen = fast_expansion_sum_zeroelim(abdlen, abd, dealen, dea, temp48b); for (i = 0; i < temp48blen; i++) { temp48b[i] = -temp48b[i]; } deablen = fast_expansion_sum_zeroelim(temp48alen, temp48a, temp48blen, temp48b, deab); xlen = scale_expansion_zeroelim(deablen, deab, pc[0], temp192); xlen = scale_expansion_zeroelim(xlen, temp192, pc[0], det384x); ylen = scale_expansion_zeroelim(deablen, deab, pc[1], temp192); ylen = scale_expansion_zeroelim(ylen, temp192, pc[1], det384y); zlen = scale_expansion_zeroelim(deablen, deab, pc[2], temp192); zlen = scale_expansion_zeroelim(zlen, temp192, pc[2], det384z); xylen = fast_expansion_sum_zeroelim(xlen, det384x, ylen, det384y, detxy); clen = fast_expansion_sum_zeroelim(xylen, detxy, zlen, det384z, cdet); temp48alen = fast_expansion_sum_zeroelim(abclen, abc, eaclen, eac, temp48a); temp48blen = fast_expansion_sum_zeroelim(bcelen, bce, eablen, eab, temp48b); for (i = 0; i < temp48blen; i++) { temp48b[i] = -temp48b[i]; } eabclen = fast_expansion_sum_zeroelim(temp48alen, temp48a, temp48blen, temp48b, eabc); xlen = scale_expansion_zeroelim(eabclen, eabc, pd[0], temp192); xlen = scale_expansion_zeroelim(xlen, temp192, pd[0], det384x); ylen = scale_expansion_zeroelim(eabclen, eabc, pd[1], temp192); ylen = scale_expansion_zeroelim(ylen, temp192, pd[1], det384y); zlen = scale_expansion_zeroelim(eabclen, eabc, pd[2], temp192); zlen = scale_expansion_zeroelim(zlen, temp192, pd[2], det384z); xylen = fast_expansion_sum_zeroelim(xlen, det384x, ylen, det384y, detxy); dlen = fast_expansion_sum_zeroelim(xylen, detxy, zlen, det384z, ddet); temp48alen = fast_expansion_sum_zeroelim(bcdlen, bcd, abdlen, abd, temp48a); temp48blen = fast_expansion_sum_zeroelim(cdalen, cda, abclen, abc, temp48b); for (i = 0; i < temp48blen; i++) { temp48b[i] = -temp48b[i]; } abcdlen = fast_expansion_sum_zeroelim(temp48alen, temp48a, temp48blen, temp48b, abcd); xlen = scale_expansion_zeroelim(abcdlen, abcd, pe[0], temp192); xlen = scale_expansion_zeroelim(xlen, temp192, pe[0], det384x); ylen = scale_expansion_zeroelim(abcdlen, abcd, pe[1], temp192); ylen = scale_expansion_zeroelim(ylen, temp192, pe[1], det384y); zlen = scale_expansion_zeroelim(abcdlen, abcd, pe[2], temp192); zlen = scale_expansion_zeroelim(zlen, temp192, pe[2], det384z); xylen = fast_expansion_sum_zeroelim(xlen, det384x, ylen, det384y, detxy); elen = fast_expansion_sum_zeroelim(xylen, detxy, zlen, det384z, edet); ablen = fast_expansion_sum_zeroelim(alen, adet, blen, bdet, abdet); cdlen = fast_expansion_sum_zeroelim(clen, cdet, dlen, ddet, cddet); cdelen = fast_expansion_sum_zeroelim(cdlen, cddet, elen, edet, cdedet); deterlen = fast_expansion_sum_zeroelim(ablen, abdet, cdelen, cdedet, deter); return deter[deterlen - 1]; } static REAL insphereadapt(const REAL *pa, const REAL *pb, const REAL *pc, const REAL *pd, const REAL *pe, REAL permanent) { INEXACT REAL aex, bex, cex, dex, aey, bey, cey, dey, aez, bez, cez, dez; REAL det, errbound; INEXACT REAL aexbey1, bexaey1, bexcey1, cexbey1; INEXACT REAL cexdey1, dexcey1, dexaey1, aexdey1; INEXACT REAL aexcey1, cexaey1, bexdey1, dexbey1; REAL aexbey0, bexaey0, bexcey0, cexbey0; REAL cexdey0, dexcey0, dexaey0, aexdey0; REAL aexcey0, cexaey0, bexdey0, dexbey0; REAL ab[4], bc[4], cd[4], da[4], ac[4], bd[4]; INEXACT REAL ab3, bc3, cd3, da3, ac3, bd3; REAL abeps, bceps, cdeps, daeps, aceps, bdeps; REAL temp8a[8], temp8b[8], temp8c[8], temp16[16], temp24[24], temp48[48]; int temp8alen, temp8blen, temp8clen, temp16len, temp24len, temp48len; REAL xdet[96], ydet[96], zdet[96], xydet[192]; int xlen, ylen, zlen, xylen; REAL adet[288], bdet[288], cdet[288], ddet[288]; int alen, blen, clen, dlen; REAL abdet[576], cddet[576]; int ablen, cdlen; REAL fin1[1152]; int finlength; REAL aextail, bextail, cextail, dextail; REAL aeytail, beytail, ceytail, deytail; REAL aeztail, beztail, ceztail, deztail; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; aex = (REAL) (pa[0] - pe[0]); bex = (REAL) (pb[0] - pe[0]); cex = (REAL) (pc[0] - pe[0]); dex = (REAL) (pd[0] - pe[0]); aey = (REAL) (pa[1] - pe[1]); bey = (REAL) (pb[1] - pe[1]); cey = (REAL) (pc[1] - pe[1]); dey = (REAL) (pd[1] - pe[1]); aez = (REAL) (pa[2] - pe[2]); bez = (REAL) (pb[2] - pe[2]); cez = (REAL) (pc[2] - pe[2]); dez = (REAL) (pd[2] - pe[2]); Two_Product(aex, bey, aexbey1, aexbey0); Two_Product(bex, aey, bexaey1, bexaey0); Two_Two_Diff(aexbey1, aexbey0, bexaey1, bexaey0, ab3, ab[2], ab[1], ab[0]); ab[3] = ab3; Two_Product(bex, cey, bexcey1, bexcey0); Two_Product(cex, bey, cexbey1, cexbey0); Two_Two_Diff(bexcey1, bexcey0, cexbey1, cexbey0, bc3, bc[2], bc[1], bc[0]); bc[3] = bc3; Two_Product(cex, dey, cexdey1, cexdey0); Two_Product(dex, cey, dexcey1, dexcey0); Two_Two_Diff(cexdey1, cexdey0, dexcey1, dexcey0, cd3, cd[2], cd[1], cd[0]); cd[3] = cd3; Two_Product(dex, aey, dexaey1, dexaey0); Two_Product(aex, dey, aexdey1, aexdey0); Two_Two_Diff(dexaey1, dexaey0, aexdey1, aexdey0, da3, da[2], da[1], da[0]); da[3] = da3; Two_Product(aex, cey, aexcey1, aexcey0); Two_Product(cex, aey, cexaey1, cexaey0); Two_Two_Diff(aexcey1, aexcey0, cexaey1, cexaey0, ac3, ac[2], ac[1], ac[0]); ac[3] = ac3; Two_Product(bex, dey, bexdey1, bexdey0); Two_Product(dex, bey, dexbey1, dexbey0); Two_Two_Diff(bexdey1, bexdey0, dexbey1, dexbey0, bd3, bd[2], bd[1], bd[0]); bd[3] = bd3; temp8alen = scale_expansion_zeroelim(4, cd, bez, temp8a); temp8blen = scale_expansion_zeroelim(4, bd, -cez, temp8b); temp8clen = scale_expansion_zeroelim(4, bc, dez, temp8c); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp24len = fast_expansion_sum_zeroelim(temp8clen, temp8c, temp16len, temp16, temp24); temp48len = scale_expansion_zeroelim(temp24len, temp24, aex, temp48); xlen = scale_expansion_zeroelim(temp48len, temp48, -aex, xdet); temp48len = scale_expansion_zeroelim(temp24len, temp24, aey, temp48); ylen = scale_expansion_zeroelim(temp48len, temp48, -aey, ydet); temp48len = scale_expansion_zeroelim(temp24len, temp24, aez, temp48); zlen = scale_expansion_zeroelim(temp48len, temp48, -aez, zdet); xylen = fast_expansion_sum_zeroelim(xlen, xdet, ylen, ydet, xydet); alen = fast_expansion_sum_zeroelim(xylen, xydet, zlen, zdet, adet); temp8alen = scale_expansion_zeroelim(4, da, cez, temp8a); temp8blen = scale_expansion_zeroelim(4, ac, dez, temp8b); temp8clen = scale_expansion_zeroelim(4, cd, aez, temp8c); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp24len = fast_expansion_sum_zeroelim(temp8clen, temp8c, temp16len, temp16, temp24); temp48len = scale_expansion_zeroelim(temp24len, temp24, bex, temp48); xlen = scale_expansion_zeroelim(temp48len, temp48, bex, xdet); temp48len = scale_expansion_zeroelim(temp24len, temp24, bey, temp48); ylen = scale_expansion_zeroelim(temp48len, temp48, bey, ydet); temp48len = scale_expansion_zeroelim(temp24len, temp24, bez, temp48); zlen = scale_expansion_zeroelim(temp48len, temp48, bez, zdet); xylen = fast_expansion_sum_zeroelim(xlen, xdet, ylen, ydet, xydet); blen = fast_expansion_sum_zeroelim(xylen, xydet, zlen, zdet, bdet); temp8alen = scale_expansion_zeroelim(4, ab, dez, temp8a); temp8blen = scale_expansion_zeroelim(4, bd, aez, temp8b); temp8clen = scale_expansion_zeroelim(4, da, bez, temp8c); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp24len = fast_expansion_sum_zeroelim(temp8clen, temp8c, temp16len, temp16, temp24); temp48len = scale_expansion_zeroelim(temp24len, temp24, cex, temp48); xlen = scale_expansion_zeroelim(temp48len, temp48, -cex, xdet); temp48len = scale_expansion_zeroelim(temp24len, temp24, cey, temp48); ylen = scale_expansion_zeroelim(temp48len, temp48, -cey, ydet); temp48len = scale_expansion_zeroelim(temp24len, temp24, cez, temp48); zlen = scale_expansion_zeroelim(temp48len, temp48, -cez, zdet); xylen = fast_expansion_sum_zeroelim(xlen, xdet, ylen, ydet, xydet); clen = fast_expansion_sum_zeroelim(xylen, xydet, zlen, zdet, cdet); temp8alen = scale_expansion_zeroelim(4, bc, aez, temp8a); temp8blen = scale_expansion_zeroelim(4, ac, -bez, temp8b); temp8clen = scale_expansion_zeroelim(4, ab, cez, temp8c); temp16len = fast_expansion_sum_zeroelim(temp8alen, temp8a, temp8blen, temp8b, temp16); temp24len = fast_expansion_sum_zeroelim(temp8clen, temp8c, temp16len, temp16, temp24); temp48len = scale_expansion_zeroelim(temp24len, temp24, dex, temp48); xlen = scale_expansion_zeroelim(temp48len, temp48, dex, xdet); temp48len = scale_expansion_zeroelim(temp24len, temp24, dey, temp48); ylen = scale_expansion_zeroelim(temp48len, temp48, dey, ydet); temp48len = scale_expansion_zeroelim(temp24len, temp24, dez, temp48); zlen = scale_expansion_zeroelim(temp48len, temp48, dez, zdet); xylen = fast_expansion_sum_zeroelim(xlen, xdet, ylen, ydet, xydet); dlen = fast_expansion_sum_zeroelim(xylen, xydet, zlen, zdet, ddet); ablen = fast_expansion_sum_zeroelim(alen, adet, blen, bdet, abdet); cdlen = fast_expansion_sum_zeroelim(clen, cdet, dlen, ddet, cddet); finlength = fast_expansion_sum_zeroelim(ablen, abdet, cdlen, cddet, fin1); det = estimate(finlength, fin1); errbound = isperrboundB * permanent; if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Diff_Tail(pa[0], pe[0], aex, aextail); Two_Diff_Tail(pa[1], pe[1], aey, aeytail); Two_Diff_Tail(pa[2], pe[2], aez, aeztail); Two_Diff_Tail(pb[0], pe[0], bex, bextail); Two_Diff_Tail(pb[1], pe[1], bey, beytail); Two_Diff_Tail(pb[2], pe[2], bez, beztail); Two_Diff_Tail(pc[0], pe[0], cex, cextail); Two_Diff_Tail(pc[1], pe[1], cey, ceytail); Two_Diff_Tail(pc[2], pe[2], cez, ceztail); Two_Diff_Tail(pd[0], pe[0], dex, dextail); Two_Diff_Tail(pd[1], pe[1], dey, deytail); Two_Diff_Tail(pd[2], pe[2], dez, deztail); if ((aextail == 0.0) && (aeytail == 0.0) && (aeztail == 0.0) && (bextail == 0.0) && (beytail == 0.0) && (beztail == 0.0) && (cextail == 0.0) && (ceytail == 0.0) && (ceztail == 0.0) && (dextail == 0.0) && (deytail == 0.0) && (deztail == 0.0)) { return det; } errbound = isperrboundC * permanent + resulterrbound * Absolute(det); abeps = (aex * beytail + bey * aextail) - (aey * bextail + bex * aeytail); bceps = (bex * ceytail + cey * bextail) - (bey * cextail + cex * beytail); cdeps = (cex * deytail + dey * cextail) - (cey * dextail + dex * ceytail); daeps = (dex * aeytail + aey * dextail) - (dey * aextail + aex * deytail); aceps = (aex * ceytail + cey * aextail) - (aey * cextail + cex * aeytail); bdeps = (bex * deytail + dey * bextail) - (bey * dextail + dex * beytail); det += (((bex * bex + bey * bey + bez * bez) * ((cez * daeps + dez * aceps + aez * cdeps) + (ceztail * da3 + deztail * ac3 + aeztail * cd3)) + (dex * dex + dey * dey + dez * dez) * ((aez * bceps - bez * aceps + cez * abeps) + (aeztail * bc3 - beztail * ac3 + ceztail * ab3))) - ((aex * aex + aey * aey + aez * aez) * ((bez * cdeps - cez * bdeps + dez * bceps) + (beztail * cd3 - ceztail * bd3 + deztail * bc3)) + (cex * cex + cey * cey + cez * cez) * ((dez * abeps + aez * bdeps + bez * daeps) + (deztail * ab3 + aeztail * bd3 + beztail * da3)))) + 2.0 * (((bex * bextail + bey * beytail + bez * beztail) * (cez * da3 + dez * ac3 + aez * cd3) + (dex * dextail + dey * deytail + dez * deztail) * (aez * bc3 - bez * ac3 + cez * ab3)) - ((aex * aextail + aey * aeytail + aez * aeztail) * (bez * cd3 - cez * bd3 + dez * bc3) + (cex * cextail + cey * ceytail + cez * ceztail) * (dez * ab3 + aez * bd3 + bez * da3))); if ((det >= errbound) || (-det >= errbound)) { return det; } return insphereexact(pa, pb, pc, pd, pe); } REAL insphere(const REAL *pa, const REAL *pb, const REAL *pc, const REAL *pd, const REAL *pe) { REAL aex, bex, cex, dex; REAL aey, bey, cey, dey; REAL aez, bez, cez, dez; REAL aexbey, bexaey, bexcey, cexbey, cexdey, dexcey, dexaey, aexdey; REAL aexcey, cexaey, bexdey, dexbey; REAL alift, blift, clift, dlift; REAL ab, bc, cd, da, ac, bd; REAL abc, bcd, cda, dab; REAL aezplus, bezplus, cezplus, dezplus; REAL aexbeyplus, bexaeyplus, bexceyplus, cexbeyplus; REAL cexdeyplus, dexceyplus, dexaeyplus, aexdeyplus; REAL aexceyplus, cexaeyplus, bexdeyplus, dexbeyplus; REAL det; REAL permanent, errbound; REAL ins; FPU_ROUND_DOUBLE; aex = pa[0] - pe[0]; bex = pb[0] - pe[0]; cex = pc[0] - pe[0]; dex = pd[0] - pe[0]; aey = pa[1] - pe[1]; bey = pb[1] - pe[1]; cey = pc[1] - pe[1]; dey = pd[1] - pe[1]; aez = pa[2] - pe[2]; bez = pb[2] - pe[2]; cez = pc[2] - pe[2]; dez = pd[2] - pe[2]; aexbey = aex * bey; bexaey = bex * aey; ab = aexbey - bexaey; bexcey = bex * cey; cexbey = cex * bey; bc = bexcey - cexbey; cexdey = cex * dey; dexcey = dex * cey; cd = cexdey - dexcey; dexaey = dex * aey; aexdey = aex * dey; da = dexaey - aexdey; aexcey = aex * cey; cexaey = cex * aey; ac = aexcey - cexaey; bexdey = bex * dey; dexbey = dex * bey; bd = bexdey - dexbey; abc = aez * bc - bez * ac + cez * ab; bcd = bez * cd - cez * bd + dez * bc; cda = cez * da + dez * ac + aez * cd; dab = dez * ab + aez * bd + bez * da; alift = aex * aex + aey * aey + aez * aez; blift = bex * bex + bey * bey + bez * bez; clift = cex * cex + cey * cey + cez * cez; dlift = dex * dex + dey * dey + dez * dez; det = (dlift * abc - clift * dab) + (blift * cda - alift * bcd); aezplus = Absolute(aez); bezplus = Absolute(bez); cezplus = Absolute(cez); dezplus = Absolute(dez); aexbeyplus = Absolute(aexbey); bexaeyplus = Absolute(bexaey); bexceyplus = Absolute(bexcey); cexbeyplus = Absolute(cexbey); cexdeyplus = Absolute(cexdey); dexceyplus = Absolute(dexcey); dexaeyplus = Absolute(dexaey); aexdeyplus = Absolute(aexdey); aexceyplus = Absolute(aexcey); cexaeyplus = Absolute(cexaey); bexdeyplus = Absolute(bexdey); dexbeyplus = Absolute(dexbey); permanent = ((cexdeyplus + dexceyplus) * bezplus + (dexbeyplus + bexdeyplus) * cezplus + (bexceyplus + cexbeyplus) * dezplus) * alift + ((dexaeyplus + aexdeyplus) * cezplus + (aexceyplus + cexaeyplus) * dezplus + (cexdeyplus + dexceyplus) * aezplus) * blift + ((aexbeyplus + bexaeyplus) * dezplus + (bexdeyplus + dexbeyplus) * aezplus + (dexaeyplus + aexdeyplus) * bezplus) * clift + ((bexceyplus + cexbeyplus) * aezplus + (cexaeyplus + aexceyplus) * bezplus + (aexbeyplus + bexaeyplus) * cezplus) * dlift; errbound = isperrboundA * permanent; if ((det > errbound) || (-det > errbound)) { FPU_RESTORE; return det; } ins = insphereadapt(pa, pb, pc, pd, pe, permanent); FPU_RESTORE; return ins; }