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