/* * CDDL HEADER START * * The contents of this file are subject to the terms of the * Common Development and Distribution License (the "License"). * You may not use this file except in compliance with the License. * * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE * or http://www.opensolaris.org/os/licensing. * See the License for the specific language governing permissions * and limitations under the License. * * When distributing Covered Code, include this CDDL HEADER in each * file and include the License file at usr/src/OPENSOLARIS.LICENSE. * If applicable, add the following below this CDDL HEADER, with the * fields enclosed by brackets "[]" replaced with your own identifying * information: Portions Copyright [yyyy] [name of copyright owner] * * CDDL HEADER END */ /* * Copyright 2011 Nexenta Systems, Inc. All rights reserved. */ /* * Copyright 2005 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ /* * __vlog: double precision vector log * * Algorithm: * * Write x = 2^n z where 1 - 2^-10 <= z < 2 - 2^-9. Let m = z * rounded to nine significant bits, so m = 1 + 2^-8 k, where * 0 <= k <= 255. Let d = z - m. Then * * log(x) = n log(2) + log(m) + log(1+(d/m)) * * Let ln2hi = log(2) rounded to a multiple of 2^-42 and ln2lo * ~ log(2) - ln2hi. From a table, obtain mh and ml such that * mh = log(m) rounded to a multiple of 2^-42 and ml ~ log(m) - * mh. From the same table, obtain rh and rl such that rh = 1/m * rounded to a multiple of 2^-10 and rl ~ 1/m - rh. For |y| <= * 2^-9, approximate log(1+y) by a polynomial y+p(y) where p(y) * := y*y*(-1/2+y*(P3+y*(P4+y*(P5+y*P6)))). Now letting s = * d*rh + d*rl in double precision, we can compute the sum above * accurately as * * (n*ln2hi + mh) + (d*rh + (d*rl + (n*ln2lo + ml) + p(s))) * * When x is subnormal, we first scale it to the normal range, * adjusting n accordingly. * * Accuracy: * * The largest error observed is less than 0.8 ulps. */ #include #ifdef _LITTLE_ENDIAN #define HI(x) *(1+(int *)&x) #define LO(x) *(unsigned *)&x #define HIWORD 1 #define LOWORD 0 #else #define HI(x) *(int *)&x #define LO(x) *(1+(unsigned *)&x) #define HIWORD 0 #define LOWORD 1 #endif #ifdef __RESTRICT #define restrict _Restrict #else #define restrict #endif static const double TBL[] = { 1.00000000000000000000e+00, 0.00000000000000000000e+00, 0.00000000000000000000e+00, 0.00000000000000000000e+00, 9.96093750000000000000e-01, 1.51994163424124515728e-05, 3.89864041562759666704e-03, 2.97263469009289512726e-14, 9.92187500000000000000e-01, 6.05620155038759681518e-05, 7.78214044203195953742e-03, 2.29894100462035112076e-14, 9.88281250000000000000e-01, 1.35738416988416988208e-04, 1.16506172200843138853e-02, -1.09039749717359319029e-13, 9.84375000000000000000e-01, 2.40384615384615397959e-04, 1.55041865359635266941e-02, 1.72745674997061065553e-15, 9.80468750000000000000e-01, 3.74161877394636028203e-04, 1.93429628432113531744e-02, -8.04185385052258635682e-14, 9.77539062500000000000e-01, -4.39825858778625927714e-04, 2.31670592816044518258e-02, -7.00735970431003565857e-14, 9.73632812500000000000e-01, -2.48782081749049442231e-04, 2.69765876983001362532e-02, -9.80605051684317662887e-14, 9.69726562500000000000e-01, -2.95928030303030311244e-05, 3.07716586667083902285e-02, 4.52981425779092882775e-14, 9.65820312500000000000e-01, 2.17423349056603779517e-04, 3.45523815067281248048e-02, -6.83913974232877736961e-14, 9.62890625000000000000e-01, -4.84609962406015010693e-04, 3.83188643020275776507e-02, 1.09021543022033016421e-13, 9.58984375000000000000e-01, -1.82876872659176042957e-04, 4.20712139207353175152e-02, -4.82631400055112824008e-14, 9.55078125000000000000e-01, 1.45755597014925360189e-04, 4.58095360313564015087e-02, -6.21983419947579227529e-14, 9.52148437500000000000e-01, -4.75575046468401500289e-04, 4.95339351223265111912e-02, -4.98803091079814255646e-14, 9.48242187500000000000e-01, -9.40393518518518520526e-05, 5.32445145188376045553e-02, -2.53216894311744497863e-14, 9.44335937500000000000e-01, 3.13508994464944631443e-04, 5.69413764001183153596e-02, 2.01093994355649575698e-14, 9.41406250000000000000e-01, -2.29779411764705879164e-04, 6.06246218164869787870e-02, -5.21362063913650408235e-14, 9.37500000000000000000e-01, 2.28937728937728937530e-04, 6.42943507054951624013e-02, -9.79051851199021608925e-14, 9.34570312500000000000e-01, -2.63743156934306572509e-04, 6.79506619085259444546e-02, -1.81950600301688149235e-14, 9.30664062500000000000e-01, 2.45028409090909096626e-04, 7.15936531869374448434e-02, 7.13730822534317801406e-14, 9.27734375000000000000e-01, -1.98143115942028998078e-04, 7.52234212375242350390e-02, 6.32906595872454402199e-14, 9.23828125000000000000e-01, 3.59600631768953083074e-04, 7.88400617077513743425e-02, 2.46501890617661192316e-14, 9.20898437500000000000e-01, -3.51281474820143869292e-05, 8.24436692109884461388e-02, 8.61451293608781447223e-14, 9.17968750000000000000e-01, -4.06025985663082419983e-04, 8.60343373417435941519e-02, 5.95592298762564263463e-14, 9.14062500000000000000e-01, 2.23214285714285707316e-04, 8.96121586897606903221e-02, -7.35577021943502867846e-14, 9.11132812500000000000e-01, -1.00784030249110321056e-04, 9.31772248541165026836e-02, 6.67870851716289831942e-14, 9.08203125000000000000e-01, -4.01706560283687926730e-04, 9.67296264584547316190e-02, 9.63806765855227740728e-14, 9.04296875000000000000e-01, 2.96764575971731443208e-04, 1.00269453163718935684e-01, -4.37863761707839790971e-14, 9.01367187500000000000e-01, 4.12632042253521119125e-05, 1.03796793681567578460e-01, 7.59863659719414144342e-14, 8.98437500000000000000e-01, 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8.71093750000000000000e-01, -3.45450680272108847594e-04, 1.38402322859064952354e-01, 5.41833313790089940464e-14, 8.68164062500000000000e-01, -3.67452330508474583805e-04, 1.41797911860294334474e-01, -3.69845950669709681858e-14, 8.65234375000000000000e-01, -3.69510135135135155647e-04, 1.45182009844575077295e-01, -7.71800133682809851086e-14, 8.62304687500000000000e-01, -3.51825547138047162871e-04, 1.48554694323138392065e-01, -1.24915489807515996540e-15, 8.59375000000000000000e-01, -3.14597315436241590364e-04, 1.51916042025732167531e-01, 1.09807540998552379211e-13, 8.56445312500000000000e-01, -2.58021530100334438914e-04, 1.55266128911080159014e-01, 4.37925082924060541938e-14, 8.53515625000000000000e-01, -1.82291666666666674979e-04, 1.58605030176659056451e-01, -2.04723578004619553937e-14, 8.50585937500000000000e-01, -8.75986295681063168849e-05, 1.61932820269385047141e-01, -7.17939001929567730476e-14, 8.47656250000000000000e-01, 2.58692052980132450107e-05, 1.65249572895390883787e-01, 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5.07812500000000000000e-01, 1.24007936507936501053e-04, 6.77398823591829568613e-01, -2.34278036379790696248e-14, 5.06835937500000000000e-01, 9.47555693069306959140e-05, 6.79380984795898257289e-01, -1.00907141981183426552e-13, 5.05859375000000000000e-01, 6.94787549407114679145e-05, 6.81359224807920327294e-01, -1.72583456150091690167e-14, 5.04882812500000000000e-01, 4.81539694280078915244e-05, 6.83333559111588328960e-01, 3.23592040115024425781e-14, 5.03906250000000000000e-01, 3.07578740157480310692e-05, 6.85304003098963221419e-01, -4.38048746232309815355e-14, 5.02929687500000000000e-01, 1.72673133595284864178e-05, 6.87270572070929119946e-01, 3.11475515031130920163e-14, 5.01953125000000000000e-01, 7.65931372549019597214e-06, 6.89233281238784911693e-01, 2.40686318405286681994e-14, 5.00976562500000000000e-01, 1.91108121330724059841e-06, 6.91192145724244255689e-01, -1.02296829368141946888e-13, }; static const double C[] = { 6.93147180559890330187e-01, 5.49792301870837115524e-14, -0.5, 3.33333333332438282293284931714682042701467889609e-0001, -2.49999999998669026809069285994497705748522309858e-0001, 2.00000758613044543658508591796951886624273250472e-0001, -1.66667492411916229281646821123333564982955309481e-0001, 4503599627370496.0, 0.0 }; #define ln2hi C[0] #define ln2lo C[1] #define mhalf C[2] #define P3 C[3] #define P4 C[4] #define P5 C[5] #define P6 C[6] #define two52 C[7] #define zero C[8] #define PROCESS(N) \ i##N = (i##N + 0x800) & ~0xfff; \ e = (i##N & 0x7ff00000) - 0x3ff00000; \ z##N.i[HIWORD] -= e; \ w##N.i[HIWORD] = i##N - e; \ w##N.i[LOWORD] = 0; \ n##N += (e >> 20); \ i##N = (i##N >> 10) & 0x3fc; \ d##N = z##N.d - w##N.d; \ h##N = d##N * TBL[i##N]; \ l##N = d##N * TBL[i##N+1]; \ s##N = h##N + l##N; \ b##N = (s##N * s##N) * (mhalf + s##N * (P3 + s##N * (P4 + \ s##N * (P5 + s##N * P6)))); \ *y = (n##N * ln2hi + TBL[i##N+2]) + (h##N + (l##N + \ (n##N * ln2lo + TBL[i##N+3]) + b##N)); \ y += stridey #define PREPROCESS(N, index, label) \ i##N = HI(*x); \ z##N.d = *x; \ x += stridex; \ n##N = 0; \ if ((i##N & 0x7ff00000) == 0x7ff00000) { /* inf or NaN */ \ y[index] = z##N.d * ((i##N < 0)? zero : z##N.d); \ goto label; \ } else if (i##N < 0x00100000) { /* subnormal, negative, zero */ \ if (((i##N << 1) | z##N.i[LOWORD]) == 0) { \ y[index] = mhalf / zero; \ goto label; \ } else if (i##N < 0) { \ y[index] = zero / zero; \ goto label; \ } \ z##N.d *= two52; \ n##N = -52; \ i##N = z##N.i[HIWORD]; \ } void __vlog(int n, double *restrict x, int stridex, double *restrict y, int stridey) { union { unsigned i[2]; double d; } z0, z1, z2, z3, w0, w1, w2, w3; double b0, b1, b2, b3; double d0, d1, d2, d3; double h0, h1, h2, h3; double l0, l1, l2, l3; double s0, s1, s2, s3; int i0, i1, i2, i3, e; int n0, n1, n2, n3; w0.i[LOWORD] = 0; w1.i[LOWORD] = 0; w2.i[LOWORD] = 0; w3.i[LOWORD] = 0; y -= stridey; for (;;) { begin: y += stridey; if (--n < 0) break; PREPROCESS(0, 0, begin); if (--n < 0) goto process1; PREPROCESS(1, stridey, process1); if (--n < 0) goto process2; PREPROCESS(2, (stridey << 1), process2); if (--n < 0) goto process3; PREPROCESS(3, (stridey << 1) + stridey, process3); i0 = (i0 + 0x800) & ~0xfff; e = (i0 & 0x7ff00000) - 0x3ff00000; z0.i[HIWORD] -= e; w0.i[HIWORD] = i0 - e; n0 += (e >> 20); i0 = (i0 >> 10) & 0x3fc; i1 = (i1 + 0x800) & ~0xfff; e = (i1 & 0x7ff00000) - 0x3ff00000; z1.i[HIWORD] -= e; w1.i[HIWORD] = i1 - e; n1 += (e >> 20); i1 = (i1 >> 10) & 0x3fc; i2 = (i2 + 0x800) & ~0xfff; e = (i2 & 0x7ff00000) - 0x3ff00000; z2.i[HIWORD] -= e; w2.i[HIWORD] = i2 - e; n2 += (e >> 20); i2 = (i2 >> 10) & 0x3fc; i3 = (i3 + 0x800) & ~0xfff; e = (i3 & 0x7ff00000) - 0x3ff00000; z3.i[HIWORD] -= e; w3.i[HIWORD] = i3 - e; n3 += (e >> 20); i3 = (i3 >> 10) & 0x3fc; d0 = z0.d - w0.d; d1 = z1.d - w1.d; d2 = z2.d - w2.d; d3 = z3.d - w3.d; h0 = d0 * TBL[i0]; h1 = d1 * TBL[i1]; h2 = d2 * TBL[i2]; h3 = d3 * TBL[i3]; l0 = d0 * TBL[i0+1]; l1 = d1 * TBL[i1+1]; l2 = d2 * TBL[i2+1]; l3 = d3 * TBL[i3+1]; s0 = h0 + l0; s1 = h1 + l1; s2 = h2 + l2; s3 = h3 + l3; b0 = (s0 * s0) * (mhalf + s0 * (P3 + s0 * (P4 + s0 * (P5 + s0 * P6)))); b1 = (s1 * s1) * (mhalf + s1 * (P3 + s1 * (P4 + s1 * (P5 + s1 * P6)))); b2 = (s2 * s2) * (mhalf + s2 * (P3 + s2 * (P4 + s2 * (P5 + s2 * P6)))); b3 = (s3 * s3) * (mhalf + s3 * (P3 + s3 * (P4 + s3 * (P5 + s3 * P6)))); *y = (n0 * ln2hi + TBL[i0+2]) + (h0 + (l0 + (n0 * ln2lo + TBL[i0+3]) + b0)); y += stridey; *y = (n1 * ln2hi + TBL[i1+2]) + (h1 + (l1 + (n1 * ln2lo + TBL[i1+3]) + b1)); y += stridey; *y = (n2 * ln2hi + TBL[i2+2]) + (h2 + (l2 + (n2 * ln2lo + TBL[i2+3]) + b2)); y += stridey; *y = (n3 * ln2hi + TBL[i3+2]) + (h3 + (l3 + (n3 * ln2lo + TBL[i3+3]) + b3)); continue; process1: PROCESS(0); continue; process2: PROCESS(0); PROCESS(1); continue; process3: PROCESS(0); PROCESS(1); PROCESS(2); } }