1/* @(#)s_expm1.c 5.1 93/09/24 */ 2/* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, 13 for performance improvement on pipelined processors. 14*/ 15 16#if defined(LIBM_SCCS) && !defined(lint) 17static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $"; 18#endif 19 20/* expm1(x) 21 * Returns exp(x)-1, the exponential of x minus 1. 22 * 23 * Method 24 * 1. Argument reduction: 25 * Given x, find r and integer k such that 26 * 27 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 28 * 29 * Here a correction term c will be computed to compensate 30 * the error in r when rounded to a floating-point number. 31 * 32 * 2. Approximating expm1(r) by a special rational function on 33 * the interval [0,0.34658]: 34 * Since 35 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 36 * we define R1(r*r) by 37 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 38 * That is, 39 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 40 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 41 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 42 * We use a special Reme algorithm on [0,0.347] to generate 43 * a polynomial of degree 5 in r*r to approximate R1. The 44 * maximum error of this polynomial approximation is bounded 45 * by 2**-61. In other words, 46 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 47 * where Q1 = -1.6666666666666567384E-2, 48 * Q2 = 3.9682539681370365873E-4, 49 * Q3 = -9.9206344733435987357E-6, 50 * Q4 = 2.5051361420808517002E-7, 51 * Q5 = -6.2843505682382617102E-9; 52 * (where z=r*r, and the values of Q1 to Q5 are listed below) 53 * with error bounded by 54 * | 5 | -61 55 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 56 * | | 57 * 58 * expm1(r) = exp(r)-1 is then computed by the following 59 * specific way which minimize the accumulation rounding error: 60 * 2 3 61 * r r [ 3 - (R1 + R1*r/2) ] 62 * expm1(r) = r + --- + --- * [--------------------] 63 * 2 2 [ 6 - r*(3 - R1*r/2) ] 64 * 65 * To compensate the error in the argument reduction, we use 66 * expm1(r+c) = expm1(r) + c + expm1(r)*c 67 * ~ expm1(r) + c + r*c 68 * Thus c+r*c will be added in as the correction terms for 69 * expm1(r+c). Now rearrange the term to avoid optimization 70 * screw up: 71 * ( 2 2 ) 72 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 73 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 74 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 75 * ( ) 76 * 77 * = r - E 78 * 3. Scale back to obtain expm1(x): 79 * From step 1, we have 80 * expm1(x) = either 2^k*[expm1(r)+1] - 1 81 * = or 2^k*[expm1(r) + (1-2^-k)] 82 * 4. Implementation notes: 83 * (A). To save one multiplication, we scale the coefficient Qi 84 * to Qi*2^i, and replace z by (x^2)/2. 85 * (B). To achieve maximum accuracy, we compute expm1(x) by 86 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 87 * (ii) if k=0, return r-E 88 * (iii) if k=-1, return 0.5*(r-E)-0.5 89 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 90 * else return 1.0+2.0*(r-E); 91 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 92 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 93 * (vii) return 2^k(1-((E+2^-k)-r)) 94 * 95 * Special cases: 96 * expm1(INF) is INF, expm1(NaN) is NaN; 97 * expm1(-INF) is -1, and 98 * for finite argument, only expm1(0)=0 is exact. 99 * 100 * Accuracy: 101 * according to an error analysis, the error is always less than 102 * 1 ulp (unit in the last place). 103 * 104 * Misc. info. 105 * For IEEE double 106 * if x > 7.09782712893383973096e+02 then expm1(x) overflow 107 * 108 * Constants: 109 * The hexadecimal values are the intended ones for the following 110 * constants. The decimal values may be used, provided that the 111 * compiler will convert from decimal to binary accurately enough 112 * to produce the hexadecimal values shown. 113 */ 114 115#include "math.h" 116#include "math_private.h" 117#define one Q[0] 118#ifdef __STDC__ 119static const double 120#else 121static double 122#endif 123huge = 1.0e+300, 124tiny = 1.0e-300, 125o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 126ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 127ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 128invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 129 /* scaled coefficients related to expm1 */ 130Q[] = {1.0, -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 131 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 132 -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 133 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 134 -2.01099218183624371326e-07}; /* BE8AFDB7 6E09C32D */ 135 136#ifdef __STDC__ 137 double __expm1(double x) 138#else 139 double __expm1(x) 140 double x; 141#endif 142{ 143 double y,hi,lo,c,t,e,hxs,hfx,r1,h2,h4,R1,R2,R3; 144 int32_t k,xsb; 145 u_int32_t hx; 146 147 GET_HIGH_WORD(hx,x); 148 xsb = hx&0x80000000; /* sign bit of x */ 149 if(xsb==0) y=x; else y= -x; /* y = |x| */ 150 hx &= 0x7fffffff; /* high word of |x| */ 151 152 /* filter out huge and non-finite argument */ 153 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 154 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 155 if(hx>=0x7ff00000) { 156 u_int32_t low; 157 GET_LOW_WORD(low,x); 158 if(((hx&0xfffff)|low)!=0) 159 return x+x; /* NaN */ 160 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 161 } 162 if(x > o_threshold) return huge*huge; /* overflow */ 163 } 164 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 165 if(x+tiny<0.0) /* raise inexact */ 166 return tiny-one; /* return -1 */ 167 } 168 } 169 170 /* argument reduction */ 171 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 172 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 173 if(xsb==0) 174 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 175 else 176 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 177 } else { 178 k = invln2*x+((xsb==0)?0.5:-0.5); 179 t = k; 180 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 181 lo = t*ln2_lo; 182 } 183 x = hi - lo; 184 c = (hi-x)-lo; 185 } 186 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 187 t = huge+x; /* return x with inexact flags when x!=0 */ 188 return x - (t-(huge+x)); 189 } 190 else k = 0; 191 192 /* x is now in primary range */ 193 hfx = 0.5*x; 194 hxs = x*hfx; 195#ifdef DO_NOT_USE_THIS 196 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 197#else 198 R1 = one+hxs*Q[1]; h2 = hxs*hxs; 199 R2 = Q[2]+hxs*Q[3]; h4 = h2*h2; 200 R3 = Q[4]+hxs*Q[5]; 201 r1 = R1 + h2*R2 + h4*R3; 202#endif 203 t = 3.0-r1*hfx; 204 e = hxs*((r1-t)/(6.0 - x*t)); 205 if(k==0) return x - (x*e-hxs); /* c is 0 */ 206 else { 207 e = (x*(e-c)-c); 208 e -= hxs; 209 if(k== -1) return 0.5*(x-e)-0.5; 210 if(k==1) { 211 if(x < -0.25) return -2.0*(e-(x+0.5)); 212 else return one+2.0*(x-e); 213 } 214 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 215 u_int32_t high; 216 y = one-(e-x); 217 GET_HIGH_WORD(high,y); 218 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 219 return y-one; 220 } 221 t = one; 222 if(k<20) { 223 u_int32_t high; 224 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ 225 y = t-(e-x); 226 GET_HIGH_WORD(high,y); 227 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 228 } else { 229 u_int32_t high; 230 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ 231 y = x-(e+t); 232 y += one; 233 GET_HIGH_WORD(high,y); 234 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 235 } 236 } 237 return y; 238} 239weak_alias (__expm1, expm1) 240#ifdef NO_LONG_DOUBLE 241strong_alias (__expm1, __expm1l) 242weak_alias (__expm1, expm1l) 243#endif 244