src/math/expm1.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.c */
   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 /* 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 Remez 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 "libm.h"
 108
 109 static const double
 110 o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
 111 ln2_hi      = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
 112 ln2_lo      = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
 113 invln2      = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
 114 /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
 115 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
 116 Q2 =  1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
 117 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
 118 Q4 =  4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
 119 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
 120
 121 double expm1(double x)
 122 {
 123         double_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
 124         union {double f; uint64_t i;} u = {x};
 125         uint32_t hx = u.i>>32 & 0x7fffffff;
 126         int k, sign = u.i>>63;
 127
 128         /* filter out huge and non-finite argument */
 129         if (hx >= 0x4043687A) {  /* if |x|>=56*ln2 */
 130                 if (isnan(x))
 131                         return x;
 132                 if (sign)
 133                         return -1;
 134                 if (x > o_threshold) {
 135                         x *= 0x1p1023;
 136                         return x;
 137                 }
 138         }
 139
 140         /* argument reduction */
 141         if (hx > 0x3fd62e42) {  /* if  |x| > 0.5 ln2 */
 142                 if (hx < 0x3FF0A2B2) {  /* and |x| < 1.5 ln2 */
 143                         if (!sign) {
 144                                 hi = x - ln2_hi;
 145                                 lo = ln2_lo;
 146                                 k =  1;
 147                         } else {
 148                                 hi = x + ln2_hi;
 149                                 lo = -ln2_lo;
 150                                 k = -1;
 151                         }
 152                 } else {
 153                         k  = invln2*x + (sign ? -0.5 : 0.5);
 154                         t  = k;
 155                         hi = x - t*ln2_hi;  /* t*ln2_hi is exact here */
 156                         lo = t*ln2_lo;
 157                 }
 158                 x = hi-lo;
 159                 c = (hi-x)-lo;
 160         } else if (hx < 0x3c900000) {  /* |x| < 2**-54, return x */
 161                 if (hx < 0x00100000)
 162                         FORCE_EVAL((float)x);
 163                 return x;
 164         } else
 165                 k = 0;
 166
 167         /* x is now in primary range */
 168         hfx = 0.5*x;
 169         hxs = x*hfx;
 170         r1 = 1.0+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
 171         t  = 3.0-r1*hfx;
 172         e  = hxs*((r1-t)/(6.0 - x*t));
 173         if (k == 0)   /* c is 0 */
 174                 return x - (x*e-hxs);
 175         e  = x*(e-c) - c;
 176         e -= hxs;
 177         /* exp(x) ~ 2^k (x_reduced - e + 1) */
 178         if (k == -1)
 179                 return 0.5*(x-e) - 0.5;
 180         if (k == 1) {
 181                 if (x < -0.25)
 182                         return -2.0*(e-(x+0.5));
 183                 return 1.0+2.0*(x-e);
 184         }
 185         u.i = (uint64_t)(0x3ff + k)<<52;  /* 2^k */
 186         twopk = u.f;
 187         if (k < 0 || k > 56) {  /* suffice to return exp(x)-1 */
 188                 y = x - e + 1.0;
 189                 if (k == 1024)
 190                         y = y*2.0*0x1p1023;
 191                 else
 192                         y = y*twopk;
 193                 return y - 1.0;
 194         }
 195         u.i = (uint64_t)(0x3ff - k)<<52;  /* 2^-k */
 196         if (k < 20)
 197                 y = (x-e+(1-u.f))*twopk;
 198         else
 199                 y = (x-(e+u.f)+1)*twopk;
 200         return y;
 201 }