1 /* derived from /netlib/fdlibm */ 2 3 /* @(#)s_expm1.c 1.3 95/01/18 */ 4 /* 5 * ==================================================== 6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 7 * 8 * Developed at SunSoft, a Sun Microsystems, Inc. business. 9 * Permission to use, copy, modify, and distribute this 10 * software is freely granted, provided that this notice 11 * is preserved. 12 * ==================================================== 13 */ 14 15 /* expm1(x) 16 * Returns exp(x)-1, the exponential of x minus 1. 17 * 18 * Method 19 * 1. Argument reduction: 20 * Given x, find r and integer k such that 21 * 22 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 23 * 24 * Here a correction term c will be computed to compensate 25 * the error in r when rounded to a floating-point number. 26 * 27 * 2. Approximating expm1(r) by a special rational function on 28 * the interval [0,0.34658]: 29 * Since 30 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 31 * we define R1(r*r) by 32 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 33 * That is, 34 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 35 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 36 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 37 * We use a special Reme algorithm on [0,0.347] to generate 38 * a polynomial of degree 5 in r*r to approximate R1. The 39 * maximum error of this polynomial approximation is bounded 40 * by 2**-61. In other words, 41 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 42 * where Q1 = -1.6666666666666567384E-2, 43 * Q2 = 3.9682539681370365873E-4, 44 * Q3 = -9.9206344733435987357E-6, 45 * Q4 = 2.5051361420808517002E-7, 46 * Q5 = -6.2843505682382617102E-9; 47 * (where z=r*r, and the values of Q1 to Q5 are listed below) 48 * with error bounded by 49 * | 5 | -61 50 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 51 * | | 52 * 53 * expm1(r) = exp(r)-1 is then computed by the following 54 * specific way which minimize the accumulation rounding error: 55 * 2 3 56 * r r [ 3 - (R1 + R1*r/2) ] 57 * expm1(r) = r + --- + --- * [--------------------] 58 * 2 2 [ 6 - r*(3 - R1*r/2) ] 59 * 60 * To compensate the error in the argument reduction, we use 61 * expm1(r+c) = expm1(r) + c + expm1(r)*c 62 * ~ expm1(r) + c + r*c 63 * Thus c+r*c will be added in as the correction terms for 64 * expm1(r+c). Now rearrange the term to avoid optimization 65 * screw up: 66 * ( 2 2 ) 67 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 68 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 69 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 70 * ( ) 71 * 72 * = r - E 73 * 3. Scale back to obtain expm1(x): 74 * From step 1, we have 75 * expm1(x) = either 2^k*[expm1(r)+1] - 1 76 * = or 2^k*[expm1(r) + (1-2^-k)] 77 * 4. Implementation notes: 78 * (A). To save one multiplication, we scale the coefficient Qi 79 * to Qi*2^i, and replace z by (x^2)/2. 80 * (B). To achieve maximum accuracy, we compute expm1(x) by 81 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 82 * (ii) if k=0, return r-E 83 * (iii) if k=-1, return 0.5*(r-E)-0.5 84 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 85 * else return 1.0+2.0*(r-E); 86 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 87 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 88 * (vii) return 2^k(1-((E+2^-k)-r)) 89 * 90 * Special cases: 91 * expm1(INF) is INF, expm1(NaN) is NaN; 92 * expm1(-INF) is -1, and 93 * for finite argument, only expm1(0)=0 is exact. 94 * 95 * Accuracy: 96 * according to an error analysis, the error is always less than 97 * 1 ulp (unit in the last place). 98 * 99 * Misc. info. 100 * For IEEE double 101 * if x > 7.09782712893383973096e+02 then expm1(x) overflow 102 * 103 * Constants: 104 * The hexadecimal values are the intended ones for the following 105 * constants. The decimal values may be used, provided that the 106 * compiler will convert from decimal to binary accurately enough 107 * to produce the hexadecimal values shown. 108 */ 109 110 #include "fdlibm.h" 111 112 static const double 113 one = 1.0, 114 Huge = 1.0e+300, 115 tiny = 1.0e-300, 116 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 117 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 118 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 119 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 120 /* scaled coefficients related to expm1 */ 121 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 122 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 123 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 124 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 125 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 126 expm1(double x)127 double expm1(double x) 128 { 129 double y,hi,lo,c,t,e,hxs,hfx,r1; 130 int k,xsb; 131 unsigned hx; 132 133 hx = __HI(x); /* high word of x */ 134 xsb = hx&0x80000000; /* sign bit of x */ 135 if(xsb==0) y=x; else y= -x; /* y = |x| */ 136 hx &= 0x7fffffff; /* high word of |x| */ 137 138 /* filter out Huge and non-finite argument */ 139 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 140 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 141 if(hx>=0x7ff00000) { 142 if(((hx&0xfffff)|__LO(x))!=0) 143 return x+x; /* NaN */ 144 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 145 } 146 if(x > o_threshold) return Huge*Huge; /* overflow */ 147 } 148 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 149 if(x+tiny<0.0) /* raise inexact */ 150 return tiny-one; /* return -1 */ 151 } 152 } 153 154 /* argument reduction */ 155 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 156 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 157 if(xsb==0) 158 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 159 else 160 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 161 } else { 162 k = invln2*x+((xsb==0)?0.5:-0.5); 163 t = k; 164 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 165 lo = t*ln2_lo; 166 } 167 x = hi - lo; 168 c = (hi-x)-lo; 169 } 170 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 171 t = Huge+x; /* return x with inexact flags when x!=0 */ 172 return x - (t-(Huge+x)); 173 } 174 else k = 0; 175 176 /* x is now in primary range */ 177 hfx = 0.5*x; 178 hxs = x*hfx; 179 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 180 t = 3.0-r1*hfx; 181 e = hxs*((r1-t)/(6.0 - x*t)); 182 if(k==0) return x - (x*e-hxs); /* c is 0 */ 183 else { 184 e = (x*(e-c)-c); 185 e -= hxs; 186 if(k== -1) return 0.5*(x-e)-0.5; 187 if(k==1) 188 if(x < -0.25) return -2.0*(e-(x+0.5)); 189 else return one+2.0*(x-e); 190 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 191 y = one-(e-x); 192 __HI(y) += (k<<20); /* add k to y's exponent */ 193 return y-one; 194 } 195 t = one; 196 if(k<20) { 197 __HI(t) = 0x3ff00000 - (0x200000>>k); /* t=1-2^-k */ 198 y = t-(e-x); 199 __HI(y) += (k<<20); /* add k to y's exponent */ 200 } else { 201 __HI(t) = ((0x3ff-k)<<20); /* 2^-k */ 202 y = x-(e+t); 203 y += one; 204 __HI(y) += (k<<20); /* add k to y's exponent */ 205 } 206 } 207 return y; 208 } 209