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 #include <sys/cdefs.h> 14 __FBSDID("$FreeBSD$"); 15 16 /* expm1(x) 17 * Returns exp(x)-1, the exponential of x minus 1. 18 * 19 * Method 20 * 1. Argument reduction: 21 * Given x, find r and integer k such that 22 * 23 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 24 * 25 * Here a correction term c will be computed to compensate 26 * the error in r when rounded to a floating-point number. 27 * 28 * 2. Approximating expm1(r) by a special rational function on 29 * the interval [0,0.34658]: 30 * Since 31 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 32 * we define R1(r*r) by 33 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 34 * That is, 35 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 36 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 37 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 38 * We use a special Reme algorithm on [0,0.347] to generate 39 * a polynomial of degree 5 in r*r to approximate R1. The 40 * maximum error of this polynomial approximation is bounded 41 * by 2**-61. In other words, 42 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 43 * where Q1 = -1.6666666666666567384E-2, 44 * Q2 = 3.9682539681370365873E-4, 45 * Q3 = -9.9206344733435987357E-6, 46 * Q4 = 2.5051361420808517002E-7, 47 * Q5 = -6.2843505682382617102E-9; 48 * z = r*r, 49 * with error bounded by 50 * | 5 | -61 51 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 52 * | | 53 * 54 * expm1(r) = exp(r)-1 is then computed by the following 55 * specific way which minimize the accumulation rounding error: 56 * 2 3 57 * r r [ 3 - (R1 + R1*r/2) ] 58 * expm1(r) = r + --- + --- * [--------------------] 59 * 2 2 [ 6 - r*(3 - R1*r/2) ] 60 * 61 * To compensate the error in the argument reduction, we use 62 * expm1(r+c) = expm1(r) + c + expm1(r)*c 63 * ~ expm1(r) + c + r*c 64 * Thus c+r*c will be added in as the correction terms for 65 * expm1(r+c). Now rearrange the term to avoid optimization 66 * screw up: 67 * ( 2 2 ) 68 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 69 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 70 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 71 * ( ) 72 * 73 * = r - E 74 * 3. Scale back to obtain expm1(x): 75 * From step 1, we have 76 * expm1(x) = either 2^k*[expm1(r)+1] - 1 77 * = or 2^k*[expm1(r) + (1-2^-k)] 78 * 4. Implementation notes: 79 * (A). To save one multiplication, we scale the coefficient Qi 80 * to Qi*2^i, and replace z by (x^2)/2. 81 * (B). To achieve maximum accuracy, we compute expm1(x) by 82 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 83 * (ii) if k=0, return r-E 84 * (iii) if k=-1, return 0.5*(r-E)-0.5 85 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 86 * else return 1.0+2.0*(r-E); 87 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 88 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 89 * (vii) return 2^k(1-((E+2^-k)-r)) 90 * 91 * Special cases: 92 * expm1(INF) is INF, expm1(NaN) is NaN; 93 * expm1(-INF) is -1, and 94 * for finite argument, only expm1(0)=0 is exact. 95 * 96 * Accuracy: 97 * according to an error analysis, the error is always less than 98 * 1 ulp (unit in the last place). 99 * 100 * Misc. info. 101 * For IEEE double 102 * if x > 7.09782712893383973096e+02 then expm1(x) overflow 103 * 104 * Constants: 105 * The hexadecimal values are the intended ones for the following 106 * constants. The decimal values may be used, provided that the 107 * compiler will convert from decimal to binary accurately enough 108 * to produce the hexadecimal values shown. 109 */ 110 111 #include "math.h" 112 #include "math_private.h" 113 114 static const double 115 one = 1.0, 116 huge = 1.0e+300, 117 tiny = 1.0e-300, 118 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 119 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 120 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 121 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 122 /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ 123 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 124 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 125 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 126 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 127 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 128 129 double 130 expm1(double x) 131 { 132 double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; 133 int32_t k,xsb; 134 u_int32_t hx; 135 136 GET_HIGH_WORD(hx,x); 137 xsb = hx&0x80000000; /* sign bit of x */ 138 if(xsb==0) y=x; else y= -x; /* y = |x| */ 139 hx &= 0x7fffffff; /* high word of |x| */ 140 141 /* filter out huge and non-finite argument */ 142 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 143 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 144 if(hx>=0x7ff00000) { 145 u_int32_t low; 146 GET_LOW_WORD(low,x); 147 if(((hx&0xfffff)|low)!=0) 148 return x+x; /* NaN */ 149 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 150 } 151 if(x > o_threshold) return huge*huge; /* overflow */ 152 } 153 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 154 if(x+tiny<0.0) /* raise inexact */ 155 return tiny-one; /* return -1 */ 156 } 157 } 158 159 /* argument reduction */ 160 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 161 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 162 if(xsb==0) 163 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 164 else 165 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 166 } else { 167 k = invln2*x+((xsb==0)?0.5:-0.5); 168 t = k; 169 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 170 lo = t*ln2_lo; 171 } 172 x = hi - lo; 173 c = (hi-x)-lo; 174 } 175 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 176 t = huge+x; /* return x with inexact flags when x!=0 */ 177 return x - (t-(huge+x)); 178 } 179 else k = 0; 180 181 /* x is now in primary range */ 182 hfx = 0.5*x; 183 hxs = x*hfx; 184 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 185 t = 3.0-r1*hfx; 186 e = hxs*((r1-t)/(6.0 - x*t)); 187 if(k==0) return x - (x*e-hxs); /* c is 0 */ 188 else { 189 INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */ 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 } 197 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 198 y = one-(e-x); 199 if (k == 1024) y = y*2.0*0x1p1023; 200 else y = y*twopk; 201 return y-one; 202 } 203 t = one; 204 if(k<20) { 205 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ 206 y = t-(e-x); 207 y = y*twopk; 208 } else { 209 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ 210 y = x-(e+t); 211 y += one; 212 y = y*twopk; 213 } 214 } 215 return y; 216 } 217