1 /* 2 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 /* expm1(x) 27 * Returns exp(x)-1, the exponential of x minus 1. 28 * 29 * Method 30 * 1. Argument reduction: 31 * Given x, find r and integer k such that 32 * 33 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 34 * 35 * Here a correction term c will be computed to compensate 36 * the error in r when rounded to a floating-point number. 37 * 38 * 2. Approximating expm1(r) by a special rational function on 39 * the interval [0,0.34658]: 40 * Since 41 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 42 * we define R1(r*r) by 43 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 44 * That is, 45 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 46 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 47 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 48 * We use a special Reme algorithm on [0,0.347] to generate 49 * a polynomial of degree 5 in r*r to approximate R1. The 50 * maximum error of this polynomial approximation is bounded 51 * by 2**-61. In other words, 52 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 53 * where Q1 = -1.6666666666666567384E-2, 54 * Q2 = 3.9682539681370365873E-4, 55 * Q3 = -9.9206344733435987357E-6, 56 * Q4 = 2.5051361420808517002E-7, 57 * Q5 = -6.2843505682382617102E-9; 58 * (where z=r*r, and the values of Q1 to Q5 are listed below) 59 * with error bounded by 60 * | 5 | -61 61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 62 * | | 63 * 64 * expm1(r) = exp(r)-1 is then computed by the following 65 * specific way which minimize the accumulation rounding error: 66 * 2 3 67 * r r [ 3 - (R1 + R1*r/2) ] 68 * expm1(r) = r + --- + --- * [--------------------] 69 * 2 2 [ 6 - r*(3 - R1*r/2) ] 70 * 71 * To compensate the error in the argument reduction, we use 72 * expm1(r+c) = expm1(r) + c + expm1(r)*c 73 * ~ expm1(r) + c + r*c 74 * Thus c+r*c will be added in as the correction terms for 75 * expm1(r+c). Now rearrange the term to avoid optimization 76 * screw up: 77 * ( 2 2 ) 78 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 79 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 80 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 81 * ( ) 82 * 83 * = r - E 84 * 3. Scale back to obtain expm1(x): 85 * From step 1, we have 86 * expm1(x) = either 2^k*[expm1(r)+1] - 1 87 * = or 2^k*[expm1(r) + (1-2^-k)] 88 * 4. Implementation notes: 89 * (A). To save one multiplication, we scale the coefficient Qi 90 * to Qi*2^i, and replace z by (x^2)/2. 91 * (B). To achieve maximum accuracy, we compute expm1(x) by 92 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 93 * (ii) if k=0, return r-E 94 * (iii) if k=-1, return 0.5*(r-E)-0.5 95 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 96 * else return 1.0+2.0*(r-E); 97 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 98 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 99 * (vii) return 2^k(1-((E+2^-k)-r)) 100 * 101 * Special cases: 102 * expm1(INF) is INF, expm1(NaN) is NaN; 103 * expm1(-INF) is -1, and 104 * for finite argument, only expm1(0)=0 is exact. 105 * 106 * Accuracy: 107 * according to an error analysis, the error is always less than 108 * 1 ulp (unit in the last place). 109 * 110 * Misc. info. 111 * For IEEE double 112 * if x > 7.09782712893383973096e+02 then expm1(x) overflow 113 * 114 * Constants: 115 * The hexadecimal values are the intended ones for the following 116 * constants. The decimal values may be used, provided that the 117 * compiler will convert from decimal to binary accurately enough 118 * to produce the hexadecimal values shown. 119 */ 120 121 #include "fdlibm.h" 122 123 #ifdef __STDC__ 124 static const double 125 #else 126 static double 127 #endif 128 one = 1.0, 129 huge = 1.0e+300, 130 tiny = 1.0e-300, 131 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 132 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 133 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 134 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 135 /* scaled coefficients related to expm1 */ 136 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 137 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 138 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 139 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 140 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 141 142 #ifdef __STDC__ expm1(double x)143 double expm1(double x) 144 #else 145 double expm1(x) 146 double x; 147 #endif 148 { 149 double y,hi,lo,c=0,t,e,hxs,hfx,r1; 150 int k,xsb; 151 unsigned hx; 152 153 hx = __HI(x); /* high word of x */ 154 xsb = hx&0x80000000; /* sign bit of x */ 155 if(xsb==0) y=x; else y= -x; /* y = |x| */ 156 hx &= 0x7fffffff; /* high word of |x| */ 157 158 /* filter out huge and non-finite argument */ 159 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 160 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 161 if(hx>=0x7ff00000) { 162 if(((hx&0xfffff)|__LO(x))!=0) 163 return x+x; /* NaN */ 164 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 165 } 166 if(x > o_threshold) return huge*huge; /* overflow */ 167 } 168 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 169 if(x+tiny<0.0) /* raise inexact */ 170 return tiny-one; /* return -1 */ 171 } 172 } 173 174 /* argument reduction */ 175 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 176 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 177 if(xsb==0) 178 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 179 else 180 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 181 } else { 182 k = invln2*x+((xsb==0)?0.5:-0.5); 183 t = k; 184 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 185 lo = t*ln2_lo; 186 } 187 x = hi - lo; 188 c = (hi-x)-lo; 189 } 190 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 191 t = huge+x; /* return x with inexact flags when x!=0 */ 192 return x - (t-(huge+x)); 193 } 194 else k = 0; 195 196 /* x is now in primary range */ 197 hfx = 0.5*x; 198 hxs = x*hfx; 199 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 200 t = 3.0-r1*hfx; 201 e = hxs*((r1-t)/(6.0 - x*t)); 202 if(k==0) return x - (x*e-hxs); /* c is 0 */ 203 else { 204 e = (x*(e-c)-c); 205 e -= hxs; 206 if(k== -1) return 0.5*(x-e)-0.5; 207 if(k==1) { 208 if(x < -0.25) return -2.0*(e-(x+0.5)); 209 else return one+2.0*(x-e); 210 } 211 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 212 y = one-(e-x); 213 __HI(y) += (k<<20); /* add k to y's exponent */ 214 return y-one; 215 } 216 t = one; 217 if(k<20) { 218 __HI(t) = 0x3ff00000 - (0x200000>>k); /* t=1-2^-k */ 219 y = t-(e-x); 220 __HI(y) += (k<<20); /* add k to y's exponent */ 221 } else { 222 __HI(t) = ((0x3ff-k)<<20); /* 2^-k */ 223 y = x-(e+t); 224 y += one; 225 __HI(y) += (k<<20); /* add k to y's exponent */ 226 } 227 } 228 return y; 229 } 230