1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* expm1(x) 13 * Returns exp(x)-1, the exponential of x minus 1. 14 * 15 * Method 16 * 1. Argument reduction: 17 * Given x, find r and integer k such that 18 * 19 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 20 * 21 * Here a correction term c will be computed to compensate 22 * the error in r when rounded to a floating-point number. 23 * 24 * 2. Approximating expm1(r) by a special rational function on 25 * the interval [0,0.34658]: 26 * Since 27 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 28 * we define R1(r*r) by 29 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 30 * That is, 31 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 32 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 33 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 34 * We use a special Reme algorithm on [0,0.347] to generate 35 * a polynomial of degree 5 in r*r to approximate R1. The 36 * maximum error of this polynomial approximation is bounded 37 * by 2**-61. In other words, 38 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 39 * where Q1 = -1.6666666666666567384E-2, 40 * Q2 = 3.9682539681370365873E-4, 41 * Q3 = -9.9206344733435987357E-6, 42 * Q4 = 2.5051361420808517002E-7, 43 * Q5 = -6.2843505682382617102E-9; 44 * z = r*r, 45 * with error bounded by 46 * | 5 | -61 47 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 48 * | | 49 * 50 * expm1(r) = exp(r)-1 is then computed by the following 51 * specific way which minimize the accumulation rounding error: 52 * 2 3 53 * r r [ 3 - (R1 + R1*r/2) ] 54 * expm1(r) = r + --- + --- * [--------------------] 55 * 2 2 [ 6 - r*(3 - R1*r/2) ] 56 * 57 * To compensate the error in the argument reduction, we use 58 * expm1(r+c) = expm1(r) + c + expm1(r)*c 59 * ~ expm1(r) + c + r*c 60 * Thus c+r*c will be added in as the correction terms for 61 * expm1(r+c). Now rearrange the term to avoid optimization 62 * screw up: 63 * ( 2 2 ) 64 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 65 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 66 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 67 * ( ) 68 * 69 * = r - E 70 * 3. Scale back to obtain expm1(x): 71 * From step 1, we have 72 * expm1(x) = either 2^k*[expm1(r)+1] - 1 73 * = or 2^k*[expm1(r) + (1-2^-k)] 74 * 4. Implementation notes: 75 * (A). To save one multiplication, we scale the coefficient Qi 76 * to Qi*2^i, and replace z by (x^2)/2. 77 * (B). To achieve maximum accuracy, we compute expm1(x) by 78 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 79 * (ii) if k=0, return r-E 80 * (iii) if k=-1, return 0.5*(r-E)-0.5 81 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 82 * else return 1.0+2.0*(r-E); 83 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 84 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 85 * (vii) return 2^k(1-((E+2^-k)-r)) 86 * 87 * Special cases: 88 * expm1(INF) is INF, expm1(NaN) is NaN; 89 * expm1(-INF) is -1, and 90 * for finite argument, only expm1(0)=0 is exact. 91 * 92 * Accuracy: 93 * according to an error analysis, the error is always less than 94 * 1 ulp (unit in the last place). 95 * 96 * Misc. info. 97 * For IEEE double 98 * if x > 7.09782712893383973096e+02 then expm1(x) overflow 99 * 100 * Constants: 101 * The hexadecimal values are the intended ones for the following 102 * constants. The decimal values may be used, provided that the 103 * compiler will convert from decimal to binary accurately enough 104 * to produce the hexadecimal values shown. 105 */ 106 107 #include <float.h> 108 109 #include "math.h" 110 #include "math_private.h" 111 112 static const double 113 one = 1.0, 114 tiny = 1.0e-300, 115 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 116 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 117 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 118 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 119 /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ 120 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 121 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 122 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 123 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 124 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 125 126 static volatile double huge = 1.0e+300; 127 128 double 129 expm1(double x) 130 { 131 double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; 132 int32_t k,xsb; 133 u_int32_t hx; 134 135 GET_HIGH_WORD(hx,x); 136 xsb = hx&0x80000000; /* sign bit of x */ 137 hx &= 0x7fffffff; /* high word of |x| */ 138 139 /* filter out huge and non-finite argument */ 140 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 141 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 142 if(hx>=0x7ff00000) { 143 u_int32_t low; 144 GET_LOW_WORD(low,x); 145 if(((hx&0xfffff)|low)!=0) 146 return x+x; /* NaN */ 147 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 148 } 149 if(x > o_threshold) return huge*huge; /* overflow */ 150 } 151 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 152 if(x+tiny<0.0) /* raise inexact */ 153 return tiny-one; /* return -1 */ 154 } 155 } 156 157 /* argument reduction */ 158 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 159 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 160 if(xsb==0) 161 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 162 else 163 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 164 } else { 165 k = invln2*x+((xsb==0)?0.5:-0.5); 166 t = k; 167 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 168 lo = t*ln2_lo; 169 } 170 STRICT_ASSIGN(double, x, hi - lo); 171 c = (hi-x)-lo; 172 } 173 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 174 t = huge+x; /* return x with inexact flags when x!=0 */ 175 return x - (t-(huge+x)); 176 } 177 else k = 0; 178 179 /* x is now in primary range */ 180 hfx = 0.5*x; 181 hxs = x*hfx; 182 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 183 t = 3.0-r1*hfx; 184 e = hxs*((r1-t)/(6.0 - x*t)); 185 if(k==0) return x - (x*e-hxs); /* c is 0 */ 186 else { 187 INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */ 188 e = (x*(e-c)-c); 189 e -= hxs; 190 if(k== -1) return 0.5*(x-e)-0.5; 191 if(k==1) { 192 if(x < -0.25) return -2.0*(e-(x+0.5)); 193 else return one+2.0*(x-e); 194 } 195 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 196 y = one-(e-x); 197 if (k == 1024) y = y*2.0*0x1p1023; 198 else y = y*twopk; 199 return y-one; 200 } 201 t = one; 202 if(k<20) { 203 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ 204 y = t-(e-x); 205 y = y*twopk; 206 } else { 207 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ 208 y = x-(e+t); 209 y += one; 210 y = y*twopk; 211 } 212 } 213 return y; 214 } 215 216 #if (LDBL_MANT_DIG == 53) 217 __weak_reference(expm1, expm1l); 218 #endif 219