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 #include <LibConfig.h> 13 #include <sys/EfiCdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $"); 16 #endif 17 18 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */ 19 // C4756: overflow in constant arithmetic 20 #pragma warning ( disable : 4756 ) 21 #endif 22 23 /* expm1(x) 24 * Returns exp(x)-1, the exponential of x minus 1. 25 * 26 * Method 27 * 1. Argument reduction: 28 * Given x, find r and integer k such that 29 * 30 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 31 * 32 * Here a correction term c will be computed to compensate 33 * the error in r when rounded to a floating-point number. 34 * 35 * 2. Approximating expm1(r) by a special rational function on 36 * the interval [0,0.34658]: 37 * Since 38 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 39 * we define R1(r*r) by 40 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 41 * That is, 42 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 43 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 44 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 45 * We use a special Reme algorithm on [0,0.347] to generate 46 * a polynomial of degree 5 in r*r to approximate R1. The 47 * maximum error of this polynomial approximation is bounded 48 * by 2**-61. In other words, 49 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 50 * where Q1 = -1.6666666666666567384E-2, 51 * Q2 = 3.9682539681370365873E-4, 52 * Q3 = -9.9206344733435987357E-6, 53 * Q4 = 2.5051361420808517002E-7, 54 * Q5 = -6.2843505682382617102E-9; 55 * (where z=r*r, and the values of Q1 to Q5 are listed below) 56 * with error bounded by 57 * | 5 | -61 58 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 59 * | | 60 * 61 * expm1(r) = exp(r)-1 is then computed by the following 62 * specific way which minimize the accumulation rounding error: 63 * 2 3 64 * r r [ 3 - (R1 + R1*r/2) ] 65 * expm1(r) = r + --- + --- * [--------------------] 66 * 2 2 [ 6 - r*(3 - R1*r/2) ] 67 * 68 * To compensate the error in the argument reduction, we use 69 * expm1(r+c) = expm1(r) + c + expm1(r)*c 70 * ~ expm1(r) + c + r*c 71 * Thus c+r*c will be added in as the correction terms for 72 * expm1(r+c). Now rearrange the term to avoid optimization 73 * screw up: 74 * ( 2 2 ) 75 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 76 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 77 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 78 * ( ) 79 * 80 * = r - E 81 * 3. Scale back to obtain expm1(x): 82 * From step 1, we have 83 * expm1(x) = either 2^k*[expm1(r)+1] - 1 84 * = or 2^k*[expm1(r) + (1-2^-k)] 85 * 4. Implementation notes: 86 * (A). To save one multiplication, we scale the coefficient Qi 87 * to Qi*2^i, and replace z by (x^2)/2. 88 * (B). To achieve maximum accuracy, we compute expm1(x) by 89 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 90 * (ii) if k=0, return r-E 91 * (iii) if k=-1, return 0.5*(r-E)-0.5 92 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 93 * else return 1.0+2.0*(r-E); 94 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 95 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 96 * (vii) return 2^k(1-((E+2^-k)-r)) 97 * 98 * Special cases: 99 * expm1(INF) is INF, expm1(NaN) is NaN; 100 * expm1(-INF) is -1, and 101 * for finite argument, only expm1(0)=0 is exact. 102 * 103 * Accuracy: 104 * according to an error analysis, the error is always less than 105 * 1 ulp (unit in the last place). 106 * 107 * Misc. info. 108 * For IEEE double 109 * if x > 7.09782712893383973096e+02 then expm1(x) overflow 110 * 111 * Constants: 112 * The hexadecimal values are the intended ones for the following 113 * constants. The decimal values may be used, provided that the 114 * compiler will convert from decimal to binary accurately enough 115 * to produce the hexadecimal values shown. 116 */ 117 118 #include "math.h" 119 #include "math_private.h" 120 121 static const double 122 one = 1.0, 123 huge = 1.0e+300, 124 tiny = 1.0e-300, 125 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 126 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 127 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 128 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 129 /* scaled coefficients related to expm1 */ 130 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 131 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 132 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 133 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 134 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 135 136 double 137 expm1(double x) 138 { 139 double y,hi,lo,c,t,e,hxs,hfx,r1; 140 int32_t k,xsb; 141 u_int32_t hx; 142 143 c = 0; 144 GET_HIGH_WORD(hx,x); 145 xsb = hx&0x80000000; /* sign bit of x */ 146 if(xsb==0) y=x; else y= -x; /* y = |x| */ 147 hx &= 0x7fffffff; /* high word of |x| */ 148 149 /* filter out huge and non-finite argument */ 150 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 151 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 152 if(hx>=0x7ff00000) { 153 u_int32_t low; 154 GET_LOW_WORD(low,x); 155 if(((hx&0xfffff)|low)!=0) 156 return x+x; /* NaN */ 157 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 158 } 159 if(x > o_threshold) return huge*huge; /* overflow */ 160 } 161 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 162 if(x+tiny<0.0) /* raise inexact */ 163 return tiny-one; /* return -1 */ 164 } 165 } 166 167 /* argument reduction */ 168 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 169 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 170 if(xsb==0) 171 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 172 else 173 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 174 } else { 175 k = (int32_t)(invln2*x+((xsb==0)?0.5:-0.5)); 176 t = k; 177 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 178 lo = t*ln2_lo; 179 } 180 x = hi - lo; 181 c = (hi-x)-lo; 182 } 183 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 184 t = huge+x; /* return x with inexact flags when x!=0 */ 185 return x - (t-(huge+x)); 186 } 187 else k = 0; 188 189 /* x is now in primary range */ 190 hfx = 0.5*x; 191 hxs = x*hfx; 192 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 193 t = 3.0-r1*hfx; 194 e = hxs*((r1-t)/(6.0 - x*t)); 195 if(k==0) return x - (x*e-hxs); /* c is 0 */ 196 else { 197 e = (x*(e-c)-c); 198 e -= hxs; 199 if(k== -1) return 0.5*(x-e)-0.5; 200 if(k==1) { 201 if(x < -0.25) return -2.0*(e-(x+0.5)); 202 else return one+2.0*(x-e); 203 } 204 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 205 u_int32_t high; 206 y = one-(e-x); 207 GET_HIGH_WORD(high,y); 208 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 209 return y-one; 210 } 211 t = one; 212 if(k<20) { 213 u_int32_t high; 214 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ 215 y = t-(e-x); 216 GET_HIGH_WORD(high,y); 217 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 218 } else { 219 u_int32_t high; 220 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ 221 y = x-(e+t); 222 y += one; 223 GET_HIGH_WORD(high,y); 224 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 225 } 226 } 227 return y; 228 } 229
