1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*- 2 * 3 * ***** BEGIN LICENSE BLOCK ***** 4 * Version: MPL 1.1/GPL 2.0/LGPL 2.1 5 * 6 * The contents of this file are subject to the Mozilla Public License Version 7 * 1.1 (the "License"); you may not use this file except in compliance with 8 * the License. You may obtain a copy of the License at 9 * http://www.mozilla.org/MPL/ 10 * 11 * Software distributed under the License is distributed on an "AS IS" basis, 12 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License 13 * for the specific language governing rights and limitations under the 14 * License. 15 * 16 * The Original Code is Mozilla Communicator client code, released 17 * March 31, 1998. 18 * 19 * The Initial Developer of the Original Code is 20 * Sun Microsystems, Inc. 21 * Portions created by the Initial Developer are Copyright (C) 1998 22 * the Initial Developer. All Rights Reserved. 23 * 24 * Contributor(s): 25 * 26 * Alternatively, the contents of this file may be used under the terms of 27 * either of the GNU General Public License Version 2 or later (the "GPL"), 28 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"), 29 * in which case the provisions of the GPL or the LGPL are applicable instead 30 * of those above. If you wish to allow use of your version of this file only 31 * under the terms of either the GPL or the LGPL, and not to allow others to 32 * use your version of this file under the terms of the MPL, indicate your 33 * decision by deleting the provisions above and replace them with the notice 34 * and other provisions required by the GPL or the LGPL. If you do not delete 35 * the provisions above, a recipient may use your version of this file under 36 * the terms of any one of the MPL, the GPL or the LGPL. 37 * 38 * ***** END LICENSE BLOCK ***** */ 39 40 /* @(#)s_expm1.c 1.3 95/01/18 */ 41 /* 42 * ==================================================== 43 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 44 * 45 * Developed at SunSoft, a Sun Microsystems, Inc. business. 46 * Permission to use, copy, modify, and distribute this 47 * software is freely granted, provided that this notice 48 * is preserved. 49 * ==================================================== 50 */ 51 52 /* expm1(x) 53 * Returns exp(x)-1, the exponential of x minus 1. 54 * 55 * Method 56 * 1. Argument reduction: 57 * Given x, find r and integer k such that 58 * 59 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 60 * 61 * Here a correction term c will be computed to compensate 62 * the error in r when rounded to a floating-point number. 63 * 64 * 2. Approximating expm1(r) by a special rational function on 65 * the interval [0,0.34658]: 66 * Since 67 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 68 * we define R1(r*r) by 69 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 70 * That is, 71 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 72 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 73 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 74 * We use a special Reme algorithm on [0,0.347] to generate 75 * a polynomial of degree 5 in r*r to approximate R1. The 76 * maximum error of this polynomial approximation is bounded 77 * by 2**-61. In other words, 78 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 79 * where Q1 = -1.6666666666666567384E-2, 80 * Q2 = 3.9682539681370365873E-4, 81 * Q3 = -9.9206344733435987357E-6, 82 * Q4 = 2.5051361420808517002E-7, 83 * Q5 = -6.2843505682382617102E-9; 84 * (where z=r*r, and the values of Q1 to Q5 are listed below) 85 * with error bounded by 86 * | 5 | -61 87 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 88 * | | 89 * 90 * expm1(r) = exp(r)-1 is then computed by the following 91 * specific way which minimize the accumulation rounding error: 92 * 2 3 93 * r r [ 3 - (R1 + R1*r/2) ] 94 * expm1(r) = r + --- + --- * [--------------------] 95 * 2 2 [ 6 - r*(3 - R1*r/2) ] 96 * 97 * To compensate the error in the argument reduction, we use 98 * expm1(r+c) = expm1(r) + c + expm1(r)*c 99 * ~ expm1(r) + c + r*c 100 * Thus c+r*c will be added in as the correction terms for 101 * expm1(r+c). Now rearrange the term to avoid optimization 102 * screw up: 103 * ( 2 2 ) 104 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 105 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 106 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 107 * ( ) 108 * 109 * = r - E 110 * 3. Scale back to obtain expm1(x): 111 * From step 1, we have 112 * expm1(x) = either 2^k*[expm1(r)+1] - 1 113 * = or 2^k*[expm1(r) + (1-2^-k)] 114 * 4. Implementation notes: 115 * (A). To save one multiplication, we scale the coefficient Qi 116 * to Qi*2^i, and replace z by (x^2)/2. 117 * (B). To achieve maximum accuracy, we compute expm1(x) by 118 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 119 * (ii) if k=0, return r-E 120 * (iii) if k=-1, return 0.5*(r-E)-0.5 121 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 122 * else return 1.0+2.0*(r-E); 123 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 124 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 125 * (vii) return 2^k(1-((E+2^-k)-r)) 126 * 127 * Special cases: 128 * expm1(INF) is INF, expm1(NaN) is NaN; 129 * expm1(-INF) is -1, and 130 * for finite argument, only expm1(0)=0 is exact. 131 * 132 * Accuracy: 133 * according to an error analysis, the error is always less than 134 * 1 ulp (unit in the last place). 135 * 136 * Misc. info. 137 * For IEEE double 138 * if x > 7.09782712893383973096e+02 then expm1(x) overflow 139 * 140 * Constants: 141 * The hexadecimal values are the intended ones for the following 142 * constants. The decimal values may be used, provided that the 143 * compiler will convert from decimal to binary accurately enough 144 * to produce the hexadecimal values shown. 145 */ 146 147 #include "fdlibm.h" 148 149 #ifdef __STDC__ 150 static const double 151 #else 152 static double 153 #endif 154 one = 1.0, 155 really_big = 1.0e+300, 156 tiny = 1.0e-300, 157 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 158 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 159 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 160 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 161 /* scaled coefficients related to expm1 */ 162 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 163 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 164 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 165 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 166 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 167 168 #ifdef __STDC__ fd_expm1(double x)169 double fd_expm1(double x) 170 #else 171 double fd_expm1(x) 172 double x; 173 #endif 174 { 175 fd_twoints u; 176 double y,hi,lo,c,t,e,hxs,hfx,r1; 177 int k,xsb; 178 unsigned hx; 179 180 u.d = x; 181 hx = __HI(u); /* high word of x */ 182 xsb = hx&0x80000000; /* sign bit of x */ 183 if(xsb==0) y=x; else y= -x; /* y = |x| */ 184 hx &= 0x7fffffff; /* high word of |x| */ 185 186 /* filter out huge and non-finite argument */ 187 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 188 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 189 if(hx>=0x7ff00000) { 190 u.d = x; 191 if(((hx&0xfffff)|__LO(u))!=0) 192 return x+x; /* NaN */ 193 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 194 } 195 if(x > o_threshold) return really_big*really_big; /* overflow */ 196 } 197 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 198 if(x+tiny<0.0) /* raise inexact */ 199 return tiny-one; /* return -1 */ 200 } 201 } 202 203 /* argument reduction */ 204 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 205 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 206 if(xsb==0) 207 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 208 else 209 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 210 } else { 211 k = (int)(invln2*x+((xsb==0)?0.5:-0.5)); 212 t = k; 213 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 214 lo = t*ln2_lo; 215 } 216 x = hi - lo; 217 c = (hi-x)-lo; 218 } 219 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 220 t = really_big+x; /* return x with inexact flags when x!=0 */ 221 return x - (t-(really_big+x)); 222 } 223 else k = 0; 224 225 /* x is now in primary range */ 226 hfx = 0.5*x; 227 hxs = x*hfx; 228 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 229 t = 3.0-r1*hfx; 230 e = hxs*((r1-t)/(6.0 - x*t)); 231 if(k==0) return x - (x*e-hxs); /* c is 0 */ 232 else { 233 e = (x*(e-c)-c); 234 e -= hxs; 235 if(k== -1) return 0.5*(x-e)-0.5; 236 if(k==1) 237 if(x < -0.25) return -2.0*(e-(x+0.5)); 238 else return one+2.0*(x-e); 239 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 240 y = one-(e-x); 241 u.d = y; 242 __HI(u) += (k<<20); /* add k to y's exponent */ 243 y = u.d; 244 return y-one; 245 } 246 t = one; 247 if(k<20) { 248 u.d = t; 249 __HI(u) = 0x3ff00000 - (0x200000>>k); /* t=1-2^-k */ 250 t = u.d; 251 y = t-(e-x); 252 u.d = y; 253 __HI(u) += (k<<20); /* add k to y's exponent */ 254 y = u.d; 255 } else { 256 u.d = t; 257 __HI(u) = ((0x3ff-k)<<20); /* 2^-k */ 258 t = u.d; 259 y = x-(e+t); 260 y += one; 261 u.d = y; 262 __HI(u) += (k<<20); /* add k to y's exponent */ 263 y = u.d; 264 } 265 } 266 return y; 267 } 268