| t2ex/bsd_source/lib/libc/src_bsd/math/s_expm1.c | bare source | permlink (0.01 seconds) |
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: /* LINTLIBRARY */ 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 Remes 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 <sys/cdefs.h> 111: #include <float.h> 112: #include <math.h> 113: 114: #include "math_private.h" 115: 116: static const double 117: one = 1.0, 118: huge = 1.0e+300, 119: tiny = 1.0e-300, 120: o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 121: ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 122: ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 123: invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 124: /* scaled coefficients related to expm1 */ 125: Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 126: Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 127: Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 128: Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 129: Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 130: 131: double 132: expm1(double x) 133: { 134: double y,hi,lo,c,t,e,hxs,hfx,r1; 135: int32_t k,xsb; 136: u_int32_t hx; 137: 138: GET_HIGH_WORD(hx,x); 139: xsb = hx&0x80000000; /* sign bit of x */ 140: if(xsb==0) y=x; else y= -x; /* y = |x| */ 141: hx &= 0x7fffffff; /* high word of |x| */ 142: 143: /* filter out huge and non-finite argument */ 144: if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 145: if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 146: if(hx>=0x7ff00000) { 147: u_int32_t low; 148: GET_LOW_WORD(low,x); 149: if(((hx&0xfffff)|low)!=0) 150: return x+x; /* NaN */ 151: else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 152: } 153: if(x > o_threshold) return huge*huge; /* overflow */ 154: } 155: if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 156: if(x+tiny<0.0) /* raise inexact */ 157: return tiny-one; /* return -1 */ 158: } 159: } 160: 161: /* argument reduction */ 162: if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 163: if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 164: if(xsb==0) 165: {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 166: else 167: {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 168: } else { 169: k = invln2*x+((xsb==0)?0.5:-0.5); 170: t = k; 171: hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 172: lo = t*ln2_lo; 173: } 174: x = hi - lo; 175: c = (hi-x)-lo; 176: } 177: else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 178: t = huge+x; /* return x with inexact flags when x!=0 */ 179: return x - (t-(huge+x)); 180: } 181: else k = 0; 182: 183: /* x is now in primary range */ 184: hfx = 0.5*x; 185: hxs = x*hfx; 186: r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 187: t = 3.0-r1*hfx; 188: e = hxs*((r1-t)/(6.0 - x*t)); 189: if(k==0) return x - (x*e-hxs); /* c is 0 */ 190: else { 191: e = (x*(e-c)-c); 192: e -= hxs; 193: if(k== -1) return 0.5*(x-e)-0.5; 194: if(k==1) 195: if(x < -0.25) return -2.0*(e-(x+0.5)); 196: else return one+2.0*(x-e); 197: if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 198: u_int32_t high; 199: y = one-(e-x); 200: GET_HIGH_WORD(high,y); 201: SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 202: return y-one; 203: } 204: t = one; 205: if(k<20) { 206: u_int32_t high; 207: SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ 208: y = t-(e-x); 209: GET_HIGH_WORD(high,y); 210: SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 211: } else { 212: u_int32_t high; 213: SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ 214: y = x-(e+t); 215: y += one; 216: GET_HIGH_WORD(high,y); 217: SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 218: } 219: } 220: return y; 221: } 222: 223: #if LDBL_MANT_DIG == 53 224: #ifdef lint 225: /* PROTOLIB1 */ 226: long double expm1l(long double); 227: #else /* lint */ 228: __weak_alias(expm1l, expm1); 229: #endif /* lint */ 230: #endif /* LDBL_MANT_DIG == 53 */