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/* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*- |
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* |
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* ***** BEGIN LICENSE BLOCK ***** |
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* Version: MPL 1.1/GPL 2.0/LGPL 2.1 |
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* |
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* The contents of this file are subject to the Mozilla Public License Version |
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* 1.1 (the "License"); you may not use this file except in compliance with |
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* the License. You may obtain a copy of the License at |
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* http://www.mozilla.org/MPL/ |
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* |
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* Software distributed under the License is distributed on an "AS IS" basis, |
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* WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License |
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* for the specific language governing rights and limitations under the |
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* License. |
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* |
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* The Original Code is Mozilla Communicator client code, released |
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* March 31, 1998. |
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* |
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* The Initial Developer of the Original Code is |
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* Sun Microsystems, Inc. |
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* Portions created by the Initial Developer are Copyright (C) 1998 |
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* the Initial Developer. All Rights Reserved. |
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* |
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* Contributor(s): |
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* |
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* Alternatively, the contents of this file may be used under the terms of |
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* either of the GNU General Public License Version 2 or later (the "GPL"), |
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* or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"), |
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* in which case the provisions of the GPL or the LGPL are applicable instead |
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* of those above. If you wish to allow use of your version of this file only |
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* under the terms of either the GPL or the LGPL, and not to allow others to |
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* use your version of this file under the terms of the MPL, indicate your |
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* decision by deleting the provisions above and replace them with the notice |
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* and other provisions required by the GPL or the LGPL. If you do not delete |
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* the provisions above, a recipient may use your version of this file under |
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* the terms of any one of the MPL, the GPL or the LGPL. |
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* |
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* ***** END LICENSE BLOCK ***** */ |
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|
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/* @(#)e_pow.c 1.3 95/01/18 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* |
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* Developed at SunSoft, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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|
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/* __ieee754_pow(x,y) return x**y |
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* |
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* n |
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* Method: Let x = 2 * (1+f) |
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* 1. Compute and return log2(x) in two pieces: |
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* log2(x) = w1 + w2, |
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* where w1 has 53-24 = 29 bit trailing zeros. |
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* 2. Perform y*log2(x) = n+y' by simulating muti-precision |
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* arithmetic, where |y'|<=0.5. |
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* 3. Return x**y = 2**n*exp(y'*log2) |
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* |
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* Special cases: |
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* 1. (anything) ** 0 is 1 |
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* 2. (anything) ** 1 is itself |
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* 3. (anything) ** NAN is NAN |
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* 4. NAN ** (anything except 0) is NAN |
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* 5. +-(|x| > 1) ** +INF is +INF |
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* 6. +-(|x| > 1) ** -INF is +0 |
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* 7. +-(|x| < 1) ** +INF is +0 |
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* 8. +-(|x| < 1) ** -INF is +INF |
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* 9. +-1 ** +-INF is NAN |
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* 10. +0 ** (+anything except 0, NAN) is +0 |
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* 11. -0 ** (+anything except 0, NAN, odd integer) is +0 |
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* 12. +0 ** (-anything except 0, NAN) is +INF |
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* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF |
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* 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) |
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* 15. +INF ** (+anything except 0,NAN) is +INF |
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* 16. +INF ** (-anything except 0,NAN) is +0 |
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* 17. -INF ** (anything) = -0 ** (-anything) |
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* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) |
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* 19. (-anything except 0 and inf) ** (non-integer) is NAN |
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* |
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* Accuracy: |
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* pow(x,y) returns x**y nearly rounded. In particular |
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* pow(integer,integer) |
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* always returns the correct integer provided it is |
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* representable. |
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* |
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* Constants : |
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* The hexadecimal values are the intended ones for the following |
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* constants. The decimal values may be used, provided that the |
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* compiler will convert from decimal to binary accurately enough |
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* to produce the hexadecimal values shown. |
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*/ |
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|
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#include "fdlibm.h" |
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|
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#if defined(_MSC_VER) |
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/* Microsoft Compiler */ |
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#pragma warning( disable : 4723 ) /* disables potential divide by 0 warning */ |
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#endif |
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|
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#ifdef __STDC__ |
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static const double |
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#else |
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static double |
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#endif |
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bp[] = {1.0, 1.5,}, |
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dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ |
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dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ |
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zero = 0.0, |
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one = 1.0, |
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two = 2.0, |
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two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ |
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really_big = 1.0e300, |
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tiny = 1.0e-300, |
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/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ |
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L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ |
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L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ |
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L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ |
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L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ |
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L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ |
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L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ |
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P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ |
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P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ |
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P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ |
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P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ |
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P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ |
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lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ |
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lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ |
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lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ |
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ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ |
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cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ |
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cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ |
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cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ |
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ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ |
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ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ |
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ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ |
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|
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#ifdef __STDC__ |
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double __ieee754_pow(double x, double y) |
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#else |
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double __ieee754_pow(x,y) |
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double x, y; |
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#endif |
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{ |
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fd_twoints ux, uy, uz; |
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double y1,t1,p_h,t,z,ax; |
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double z_h,z_l,p_l; |
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double t2,r,s,u,v,w; |
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int i,j,k,yisint,n; |
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int hx,hy,ix,iy; |
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unsigned lx,ly; |
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|
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ux.d = x; uy.d = y; |
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hx = __HI(ux); lx = __LO(ux); |
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hy = __HI(uy); ly = __LO(uy); |
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ix = hx&0x7fffffff; iy = hy&0x7fffffff; |
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|
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/* y==zero: x**0 = 1 */ |
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if((iy|ly)==0) return one; |
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|
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/* +-NaN return x+y */ |
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if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || |
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iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) |
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return x+y; |
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|
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/* determine if y is an odd int when x < 0 |
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* yisint = 0 ... y is not an integer |
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* yisint = 1 ... y is an odd int |
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* yisint = 2 ... y is an even int |
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*/ |
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yisint = 0; |
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if(hx<0) { |
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if(iy>=0x43400000) yisint = 2; /* even integer y */ |
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else if(iy>=0x3ff00000) { |
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k = (iy>>20)-0x3ff; /* exponent */ |
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if(k>20) { |
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j = ly>>(52-k); |
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if((j<<(52-k))==(int)ly) yisint = 2-(j&1); |
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} else if(ly==0) { |
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j = iy>>(20-k); |
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if((j<<(20-k))==iy) yisint = 2-(j&1); |
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} |
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} |
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} |
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|
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/* special value of y */ |
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if(ly==0) { |
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if (iy==0x7ff00000) { /* y is +-inf */ |
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if(((ix-0x3ff00000)|lx)==0) |
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#ifdef _WIN32 |
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/* VC++ optimizer reduces y - y to 0 */ |
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return y / y; |
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#else |
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return y - y; /* inf**+-1 is NaN */ |
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#endif |
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else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ |
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return (hy>=0)? y: zero; |
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else /* (|x|<1)**-,+inf = inf,0 */ |
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return (hy<0)?-y: zero; |
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} |
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if(iy==0x3ff00000) { /* y is +-1 */ |
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if(hy<0) return one/x; else return x; |
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} |
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if(hy==0x40000000) return x*x; /* y is 2 */ |
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if(hy==0x3fe00000) { /* y is 0.5 */ |
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if(hx>=0) /* x >= +0 */ |
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return fd_sqrt(x); |
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} |
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} |
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|
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ax = fd_fabs(x); |
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/* special value of x */ |
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if(lx==0) { |
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if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ |
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z = ax; /*x is +-0,+-inf,+-1*/ |
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if(hy<0) z = one/z; /* z = (1/|x|) */ |
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if(hx<0) { |
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if(((ix-0x3ff00000)|yisint)==0) { |
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z = (z-z)/(z-z); /* (-1)**non-int is NaN */ |
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} else if(yisint==1) { |
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#ifdef HPUX |
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uz.d = z; |
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__HI(uz) ^= 1<<31; /* some HPUXes cannot negate 0.. */ |
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z = uz.d; |
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#else |
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z = -z; /* (x<0)**odd = -(|x|**odd) */ |
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#endif |
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} |
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} |
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return z; |
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} |
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} |
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|
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/* (x<0)**(non-int) is NaN */ |
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if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x); |
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|
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/* |y| is really_big */ |
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if(iy>0x41e00000) { /* if |y| > 2**31 */ |
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if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ |
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if(ix<=0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny; |
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if(ix>=0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny; |
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} |
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/* over/underflow if x is not close to one */ |
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if(ix<0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny; |
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if(ix>0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny; |
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/* now |1-x| is tiny <= 2**-20, suffice to compute |
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log(x) by x-x^2/2+x^3/3-x^4/4 */ |
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t = x-1; /* t has 20 trailing zeros */ |
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w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); |
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u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ |
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v = t*ivln2_l-w*ivln2; |
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t1 = u+v; |
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uz.d = t1; |
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__LO(uz) = 0; |
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t1 = uz.d; |
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t2 = v-(t1-u); |
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} else { |
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double s_h,t_h; |
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double s2,s_l,t_l; |
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n = 0; |
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/* take care subnormal number */ |
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if(ix<0x00100000) |
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{ax *= two53; n -= 53; uz.d = ax; ix = __HI(uz); } |
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n += ((ix)>>20)-0x3ff; |
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j = ix&0x000fffff; |
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/* determine interval */ |
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ix = j|0x3ff00000; /* normalize ix */ |
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if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ |
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else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ |
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else {k=0;n+=1;ix -= 0x00100000;} |
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uz.d = ax; |
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__HI(uz) = ix; |
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ax = uz.d; |
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|
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/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
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u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ |
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v = one/(ax+bp[k]); |
| 281 |
s = u*v; |
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s_h = s; |
| 283 |
uz.d = s_h; |
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__LO(uz) = 0; |
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s_h = uz.d; |
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/* t_h=ax+bp[k] High */ |
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t_h = zero; |
| 288 |
uz.d = t_h; |
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__HI(uz)=((ix>>1)|0x20000000)+0x00080000+(k<<18); |
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t_h = uz.d; |
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t_l = ax - (t_h-bp[k]); |
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s_l = v*((u-s_h*t_h)-s_h*t_l); |
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/* compute log(ax) */ |
| 294 |
s2 = s*s; |
| 295 |
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); |
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r += s_l*(s_h+s); |
| 297 |
s2 = s_h*s_h; |
| 298 |
t_h = 3.0+s2+r; |
| 299 |
uz.d = t_h; |
| 300 |
__LO(uz) = 0; |
| 301 |
t_h = uz.d; |
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t_l = r-((t_h-3.0)-s2); |
| 303 |
/* u+v = s*(1+...) */ |
| 304 |
u = s_h*t_h; |
| 305 |
v = s_l*t_h+t_l*s; |
| 306 |
/* 2/(3log2)*(s+...) */ |
| 307 |
p_h = u+v; |
| 308 |
uz.d = p_h; |
| 309 |
__LO(uz) = 0; |
| 310 |
p_h = uz.d; |
| 311 |
p_l = v-(p_h-u); |
| 312 |
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ |
| 313 |
z_l = cp_l*p_h+p_l*cp+dp_l[k]; |
| 314 |
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
| 315 |
t = (double)n; |
| 316 |
t1 = (((z_h+z_l)+dp_h[k])+t); |
| 317 |
uz.d = t1; |
| 318 |
__LO(uz) = 0; |
| 319 |
t1 = uz.d; |
| 320 |
t2 = z_l-(((t1-t)-dp_h[k])-z_h); |
| 321 |
} |
| 322 |
|
| 323 |
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ |
| 324 |
if((((hx>>31)+1)|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ |
| 325 |
|
| 326 |
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ |
| 327 |
y1 = y; |
| 328 |
uy.d = y1; |
| 329 |
__LO(uy) = 0; |
| 330 |
y1 = uy.d; |
| 331 |
p_l = (y-y1)*t1+y*t2; |
| 332 |
p_h = y1*t1; |
| 333 |
z = p_l+p_h; |
| 334 |
uz.d = z; |
| 335 |
j = __HI(uz); |
| 336 |
i = __LO(uz); |
| 337 |
|
| 338 |
if (j>=0x40900000) { /* z >= 1024 */ |
| 339 |
if(((j-0x40900000)|i)!=0) /* if z > 1024 */ |
| 340 |
return s*really_big*really_big; /* overflow */ |
| 341 |
else { |
| 342 |
if(p_l+ovt>z-p_h) return s*really_big*really_big; /* overflow */ |
| 343 |
} |
| 344 |
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ |
| 345 |
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ |
| 346 |
return s*tiny*tiny; /* underflow */ |
| 347 |
else { |
| 348 |
if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ |
| 349 |
} |
| 350 |
} |
| 351 |
/* |
| 352 |
* compute 2**(p_h+p_l) |
| 353 |
*/ |
| 354 |
i = j&0x7fffffff; |
| 355 |
k = (i>>20)-0x3ff; |
| 356 |
n = 0; |
| 357 |
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ |
| 358 |
n = j+(0x00100000>>(k+1)); |
| 359 |
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ |
| 360 |
t = zero; |
| 361 |
uz.d = t; |
| 362 |
__HI(uz) = (n&~(0x000fffff>>k)); |
| 363 |
t = uz.d; |
| 364 |
n = ((n&0x000fffff)|0x00100000)>>(20-k); |
| 365 |
if(j<0) n = -n; |
| 366 |
p_h -= t; |
| 367 |
} |
| 368 |
t = p_l+p_h; |
| 369 |
uz.d = t; |
| 370 |
__LO(uz) = 0; |
| 371 |
t = uz.d; |
| 372 |
u = t*lg2_h; |
| 373 |
v = (p_l-(t-p_h))*lg2+t*lg2_l; |
| 374 |
z = u+v; |
| 375 |
w = v-(z-u); |
| 376 |
t = z*z; |
| 377 |
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); |
| 378 |
r = (z*t1)/(t1-two)-(w+z*w); |
| 379 |
z = one-(r-z); |
| 380 |
uz.d = z; |
| 381 |
j = __HI(uz); |
| 382 |
j += (n<<20); |
| 383 |
if((j>>20)<=0) z = fd_scalbn(z,n); /* subnormal output */ |
| 384 |
else { uz.d = z; __HI(uz) += (n<<20); z = uz.d; } |
| 385 |
return s*z; |
| 386 |
} |
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