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    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 */