/*****************************************************************************/ /* */ /* Routines for Arbitrary Precision Floating-point Arithmetic */ /* and Fast Robust Geometric Predicates */ /* (predicates.cpp) */ /* */ /* 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 */ /* */ /* incircle(pa, pb, pc, pd) */ /* incirclefast(pa, pb, pc, pd) */ /* */ /* 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 */ /* */ /* 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 QT_WARNING_PUSH QT_WARNING_DISABLE_MSVC(4701) // cppcheck-suppress unknownMacro QT_WARNING_DISABLE_GCC("-Wold-style-cast") QT_WARNING_DISABLE_GCC("-Wfloat-equal") #if defined(__GNUC__) && (__GNUC__ * 100 + __GNUC_MINOR__) >= 408 QT_WARNING_DISABLE_GCC("-Wmaybe-uninitialized") #else QT_WARNING_DISABLE_GCC("-Wuninitialized") #endif QT_WARNING_DISABLE_CLANG("-Wold-style-cast") QT_WARNING_DISABLE_CLANG("-Wmissing-variable-declarations") QT_WARNING_DISABLE_CLANG("-Wfloat-equal") QT_WARNING_DISABLE_CLANG("-Wmissing-prototypes") QT_WARNING_DISABLE_CLANG("-Wconditional-uninitialized") /* 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 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 = (qreal)(a + b); \ Fast_Two_Sum_Tail(a, b, x, y) #define Two_Sum_Tail(a, b, x, y) \ bvirt = (qreal)(x - a); \ avirt = x - bvirt; \ bround = b - bvirt; \ around = a - avirt; \ y = around + bround #define Two_Sum(a, b, x, y) \ x = (qreal)(a + b); \ Two_Sum_Tail(a, b, x, y) #define Two_Diff_Tail(a, b, x, y) \ bvirt = (qreal)(a - x); \ avirt = x + bvirt; \ bround = bvirt - b; \ around = a - avirt; \ y = around + bround #define Two_Diff(a, b, x, y) \ x = (qreal)(a - b); \ Two_Diff_Tail(a, b, x, y) #define Split(a, ahi, alo) \ c = (qreal)(splitter * a); \ abig = (qreal)(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 = (qreal)(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 = (qreal)(a * b); \ Split(a, ahi, alo); \ 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 = (qreal)(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) qreal splitter; /* = 2^ceiling(p / 2) + 1. Used to split floats in half. */ qreal epsilon; /* = 2^(-p). Used to estimate roundoff errors. */ /* A set of coefficients used to calculate maximum roundoff errors. */ qreal resulterrbound; qreal ccwerrboundA, ccwerrboundB, ccwerrboundC; qreal o3derrboundA, o3derrboundB, o3derrboundC; qreal iccerrboundA, iccerrboundB, iccerrboundC; qreal isperrboundA, isperrboundB, isperrboundC; /*****************************************************************************/ /* */ /* 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 exactinit() { qreal half; qreal 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; } /*****************************************************************************/ /* */ /* 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. */ /* */ /*****************************************************************************/ auto fast_expansion_sum_zeroelim(int elen, qreal *e, int flen, qreal *f, qreal *h) -> int /* h cannot be e or f. */ { qreal Q; INEXACT qreal Qnew; INEXACT qreal hh; INEXACT qreal bvirt; qreal avirt, bround, around; int eindex, findex, hindex; qreal 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.) */ /* */ /*****************************************************************************/ auto scale_expansion_zeroelim(int elen, qreal *e, qreal b, qreal *h) -> int /* e and h cannot be the same. */ { INEXACT qreal Q, sum; qreal hh; INEXACT qreal product1; qreal product0; int eindex, hindex; qreal enow; INEXACT qreal bvirt; qreal avirt, bround, around; INEXACT qreal c; INEXACT qreal abig; qreal ahi, alo, bhi, blo; qreal 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. */ /* */ /*****************************************************************************/ auto estimate(int elen, qreal *e) -> qreal { qreal Q; int eindex; Q = e[0]; for (eindex = 1; eindex < elen; eindex++) { Q += e[eindex]; } return Q; } auto incircleadapt(qreal *pa, qreal *pb, qreal *pc, qreal *pd, qreal permanent) -> qreal { INEXACT qreal adx, bdx, cdx, ady, bdy, cdy; qreal det, errbound; INEXACT qreal bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; qreal bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; qreal bc[4], ca[4], ab[4]; INEXACT qreal bc3, ca3, ab3; qreal axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32]; int axbclen, axxbclen, aybclen, ayybclen, alen; qreal bxca[8], bxxca[16], byca[8], byyca[16], bdet[32]; int bxcalen, bxxcalen, bycalen, byycalen, blen; qreal cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32]; int cxablen, cxxablen, cyablen, cyyablen, clen; qreal abdet[64]; int ablen; qreal fin1[1152], fin2[1152]; qreal *finnow, *finother, *finswap; int finlength; qreal adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail; INEXACT qreal adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1; qreal adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0; qreal aa[4], bb[4], cc[4]; INEXACT qreal aa3, bb3, cc3; INEXACT qreal ti1, tj1; qreal ti0, tj0; qreal u[4], v[4]; INEXACT qreal u3, v3; qreal temp8[8], temp16a[16], temp16b[16], temp16c[16]; qreal temp32a[32], temp32b[32], temp48[48], temp64[64]; int temp8len, temp16alen, temp16blen, temp16clen; int temp32alen, temp32blen, temp48len, temp64len; qreal axtbb[8], axtcc[8], aytbb[8], aytcc[8]; // cppcheck-suppress variableScope int axtbblen, axtcclen, aytbblen, aytcclen; qreal bxtaa[8], bxtcc[8], bytaa[8], bytcc[8]; // cppcheck-suppress variableScope int bxtaalen, bxtcclen, bytaalen, bytcclen; qreal cxtaa[8], cxtbb[8], cytaa[8], cytbb[8]; // cppcheck-suppress variableScope int cxtaalen, cxtbblen, cytaalen, cytbblen; qreal axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8]; int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen; qreal axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16]; // cppcheck-suppress variableScope int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen; qreal axtbctt[8], aytbctt[8], bxtcatt[8]; qreal bytcatt[8], cxtabtt[8], cytabtt[8]; // cppcheck-suppress variableScope int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen; qreal abt[8], bct[8], cat[8]; // cppcheck-suppress variableScope int abtlen, bctlen, catlen; qreal abtt[4], bctt[4], catt[4]; // cppcheck-suppress variableScope int abttlen, bcttlen, cattlen; INEXACT qreal abtt3, bctt3, catt3; qreal negate; INEXACT qreal bvirt; qreal avirt, bround, around; INEXACT qreal c; INEXACT qreal abig; qreal ahi, alo, bhi, blo; qreal err1, err2, err3; INEXACT qreal _i, _j; qreal _0; adx = (qreal)(pa[0] - pd[0]); bdx = (qreal)(pb[0] - pd[0]); cdx = (qreal)(pc[0] - pd[0]); ady = (qreal)(pa[1] - pd[1]); bdy = (qreal)(pb[1] - pd[1]); cdy = (qreal)(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]; } auto incircle(qreal *pa, qreal *pb, qreal *pc, qreal *pd) -> qreal { qreal adx, bdx, cdx, ady, bdy, cdy; qreal bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; qreal alift, blift, clift; qreal det; qreal permanent, errbound; 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)) { return det; } return incircleadapt(pa, pb, pc, pd, permanent); } QT_WARNING_POP