StdLib/LibC/Math/s_expm1.c

   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 #include  <LibConfig.h>
  13 #include  <sys/EfiCdefs.h>
  14 #if defined(LIBM_SCCS) && !defined(lint)
  15 __RCSID("$NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $");
  16 #endif
  17
  18 #if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
  19   // C4756: overflow in constant arithmetic
  20   #pragma warning ( disable : 4756 )
  21 #endif
  22
  23 /* expm1(x)
  24  * Returns exp(x)-1, the exponential of x minus 1.
  25  *
  26  * Method
  27  *   1. Argument reduction:
  28  *  Given x, find r and integer k such that
  29  *
  30  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
  31  *
  32  *      Here a correction term c will be computed to compensate
  33  *  the error in r when rounded to a floating-point number.
  34  *
  35  *   2. Approximating expm1(r) by a special rational function on
  36  *  the interval [0,0.34658]:
  37  *  Since
  38  *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
  39  *  we define R1(r*r) by
  40  *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
  41  *  That is,
  42  *      R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
  43  *         = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
  44  *         = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
  45  *      We use a special Reme algorithm on [0,0.347] to generate
  46  *  a polynomial of degree 5 in r*r to approximate R1. The
  47  *  maximum error of this polynomial approximation is bounded
  48  *  by 2**-61. In other words,
  49  *      R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
  50  *  where   Q1  =  -1.6666666666666567384E-2,
  51  *    Q2  =   3.9682539681370365873E-4,
  52  *    Q3  =  -9.9206344733435987357E-6,
  53  *    Q4  =   2.5051361420808517002E-7,
  54  *    Q5  =  -6.2843505682382617102E-9;
  55  *    (where z=r*r, and the values of Q1 to Q5 are listed below)
  56  *  with error bounded by
  57  *      |                  5           |     -61
  58  *      | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
  59  *      |                              |
  60  *
  61  *  expm1(r) = exp(r)-1 is then computed by the following
  62  *  specific way which minimize the accumulation rounding error:
  63  *             2     3
  64  *            r     r    [ 3 - (R1 + R1*r/2)  ]
  65  *        expm1(r) = r + --- + --- * [--------------------]
  66  *                  2     2    [ 6 - r*(3 - R1*r/2) ]
  67  *
  68  *  To compensate the error in the argument reduction, we use
  69  *    expm1(r+c) = expm1(r) + c + expm1(r)*c
  70  *         ~ expm1(r) + c + r*c
  71  *  Thus c+r*c will be added in as the correction terms for
  72  *  expm1(r+c). Now rearrange the term to avoid optimization
  73  *  screw up:
  74  *            (      2                                    2 )
  75  *            ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
  76  *   expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
  77  *                  ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
  78  *                      (                                             )
  79  *
  80  *       = r - E
  81  *   3. Scale back to obtain expm1(x):
  82  *  From step 1, we have
  83  *     expm1(x) = either 2^k*[expm1(r)+1] - 1
  84  *        = or     2^k*[expm1(r) + (1-2^-k)]
  85  *   4. Implementation notes:
  86  *  (A). To save one multiplication, we scale the coefficient Qi
  87  *       to Qi*2^i, and replace z by (x^2)/2.
  88  *  (B). To achieve maximum accuracy, we compute expm1(x) by
  89  *    (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
  90  *    (ii)  if k=0, return r-E
  91  *    (iii) if k=-1, return 0.5*(r-E)-0.5
  92  *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
  93  *                 else      return  1.0+2.0*(r-E);
  94  *    (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
  95  *    (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
  96  *    (vii) return 2^k(1-((E+2^-k)-r))
  97  *
  98  * Special cases:
  99  *  expm1(INF) is INF, expm1(NaN) is NaN;
 100  *  expm1(-INF) is -1, and
 101  *  for finite argument, only expm1(0)=0 is exact.
 102  *
 103  * Accuracy:
 104  *  according to an error analysis, the error is always less than
 105  *  1 ulp (unit in the last place).
 106  *
 107  * Misc. info.
 108  *  For IEEE double
 109  *      if x >  7.09782712893383973096e+02 then expm1(x) overflow
 110  *
 111  * Constants:
 112  * The hexadecimal values are the intended ones for the following
 113  * constants. The decimal values may be used, provided that the
 114  * compiler will convert from decimal to binary accurately enough
 115  * to produce the hexadecimal values shown.
 116  */
 117
 118 #include "math.h"
 119 #include "math_private.h"
 120
 121 static const double
 122 one   = 1.0,
 123 huge    = 1.0e+300,
 124 tiny    = 1.0e-300,
 125 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
 126 ln2_hi    = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
 127 ln2_lo    = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
 128 invln2    = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
 129   /* scaled coefficients related to expm1 */
 130 Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
 131 Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
 132 Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
 133 Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
 134 Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
 135
 136 double
 137 expm1(double x)
 138 {
 139   double y,hi,lo,c,t,e,hxs,hfx,r1;
 140   int32_t k,xsb;
 141   u_int32_t hx;
 142
 143   c = 0;
 144   GET_HIGH_WORD(hx,x);
 145   xsb = hx&0x80000000;    /* sign bit of x */
 146   if(xsb==0) y=x; else y= -x; /* y = |x| */
 147   hx &= 0x7fffffff;   /* high word of |x| */
 148
 149     /* filter out huge and non-finite argument */
 150   if(hx >= 0x4043687A) {      /* if |x|>=56*ln2 */
 151       if(hx >= 0x40862E42) {    /* if |x|>=709.78... */
 152                 if(hx>=0x7ff00000) {
 153         u_int32_t low;
 154         GET_LOW_WORD(low,x);
 155         if(((hx&0xfffff)|low)!=0)
 156              return x+x;   /* NaN */
 157         else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
 158           }
 159           if(x > o_threshold) return huge*huge; /* overflow */
 160       }
 161       if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
 162     if(x+tiny<0.0)    /* raise inexact */
 163     return tiny-one;  /* return -1 */
 164       }
 165   }
 166
 167     /* argument reduction */
 168   if(hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
 169       if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
 170     if(xsb==0)
 171         {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
 172     else
 173         {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
 174       } else {
 175     k  = (int32_t)(invln2*x+((xsb==0)?0.5:-0.5));
 176     t  = k;
 177     hi = x - t*ln2_hi;  /* t*ln2_hi is exact here */
 178     lo = t*ln2_lo;
 179       }
 180       x  = hi - lo;
 181       c  = (hi-x)-lo;
 182   }
 183   else if(hx < 0x3c900000) {    /* when |x|<2**-54, return x */
 184       t = huge+x; /* return x with inexact flags when x!=0 */
 185       return x - (t-(huge+x));
 186   }
 187   else k = 0;
 188
 189     /* x is now in primary range */
 190   hfx = 0.5*x;
 191   hxs = x*hfx;
 192   r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
 193   t  = 3.0-r1*hfx;
 194   e  = hxs*((r1-t)/(6.0 - x*t));
 195   if(k==0) return x - (x*e-hxs);    /* c is 0 */
 196   else {
 197       e  = (x*(e-c)-c);
 198       e -= hxs;
 199       if(k== -1) return 0.5*(x-e)-0.5;
 200       if(k==1)  {
 201           if(x < -0.25) return -2.0*(e-(x+0.5));
 202           else        return  one+2.0*(x-e);
 203       }
 204       if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
 205           u_int32_t high;
 206           y = one-(e-x);
 207     GET_HIGH_WORD(high,y);
 208     SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 209           return y-one;
 210       }
 211       t = one;
 212       if(k<20) {
 213           u_int32_t high;
 214           SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
 215           y = t-(e-x);
 216     GET_HIGH_WORD(high,y);
 217     SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 218      } else {
 219           u_int32_t high;
 220     SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
 221           y = x-(e+t);
 222           y += one;
 223     GET_HIGH_WORD(high,y);
 224     SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 225       }
 226   }
 227   return y;
 228 }