```5261 libm should stop using synonyms.h
5298 fabs is 0-sized, confuses dis(1) and others
Reviewed by: Josef 'Jeff' Sipek <jeffpc@josefsipek.net>
Approved by: Gordon Ross <gwr@nexenta.com>```

```   1 /*
3  *
4  * The contents of this file are subject to the terms of the
6  * You may not use this file except in compliance with the License.
7  *
8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9  * or http://www.opensolaris.org/os/licensing.
10  * See the License for the specific language governing permissions
11  * and limitations under the License.
12  *
13  * When distributing Covered Code, include this CDDL HEADER in each
15  * If applicable, add the following below this CDDL HEADER, with the
16  * fields enclosed by brackets "[]" replaced with your own identifying
18  *
20  */
21
22 /*
24  */
25 /*
27  * Use is subject to license terms.
28  */
29
30 #pragma weak expm1 = __expm1
31
32 /* INDENT OFF */
33 /*
34  * expm1(x)
35  * Returns exp(x)-1, the exponential of x minus 1.
36  *
37  * Method
38  *   1. Arugment reduction:
39  *      Given x, find r and integer k such that
40  *
41  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
42  *
43  *      Here a correction term c will be computed to compensate
44  *      the error in r when rounded to a floating-point number.
45  *
46  *   2. Approximating expm1(r) by a special rational function on
47  *      the interval [0,0.34658]:
48  *      Since
49  *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
50  *      we define R1(r*r) by
51  *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
52  *      That is,
53  *          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
54  *                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
55  *                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
56  *      We use a special Reme algorithm on [0,0.347] to generate
57  *      a polynomial of degree 5 in r*r to approximate R1. The
58  *      maximum error of this polynomial approximation is bounded
59  *      by 2**-61. In other words,
60  *          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
61  *      where   Q1  =  -1.6666666666666567384E-2,
62  *              Q2  =   3.9682539681370365873E-4,
63  *              Q3  =  -9.9206344733435987357E-6,
64  *              Q4  =   2.5051361420808517002E-7,
65  *              Q5  =  -6.2843505682382617102E-9;
66  *      (where z=r*r, and the values of Q1 to Q5 are listed below)
67  *      with error bounded by
68  *          |                  5           |     -61
69  *          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
70  *          |                              |
71  *
72  *      expm1(r) = exp(r)-1 is then computed by the following
73  *      specific way which minimize the accumulation rounding error:
74  *                             2     3
75  *                            r     r    [ 3 - (R1 + R1*r/2)  ]
76  *            expm1(r) = r + --- + --- * [--------------------]
77  *                            2     2    [ 6 - r*(3 - R1*r/2) ]
78  *
79  *      To compensate the error in the argument reduction, we use
80  *              expm1(r+c) = expm1(r) + c + expm1(r)*c
81  *                         ~ expm1(r) + c + r*c
82  *      Thus c+r*c will be added in as the correction terms for
83  *      expm1(r+c). Now rearrange the term to avoid optimization
84  *      screw up:
85  *                      (      2                                    2 )
86  *                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
87  *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
88  *                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
89  *                      (                                             )
90  *
91  *                 = r - E
92  *   3. Scale back to obtain expm1(x):
93  *      From step 1, we have
94  *         expm1(x) = either 2^k*[expm1(r)+1] - 1
95  *                  = or     2^k*[expm1(r) + (1-2^-k)]
96  *   4. Implementation notes:
97  *      (A). To save one multiplication, we scale the coefficient Qi
98  *           to Qi*2^i, and replace z by (x^2)/2.
99  *      (B). To achieve maximum accuracy, we compute expm1(x) by
100  *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x != inf)
101  *        (ii)  if k=0, return r-E
102  *        (iii) if k=-1, return 0.5*(r-E)-0.5
103  *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
104  *                                      else         return  1.0+2.0*(r-E);
105  *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
106  *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
107  *        (vii) return 2^k(1-((E+2^-k)-r))
108  *
109  * Special cases:
110  *      expm1(INF) is INF, expm1(NaN) is NaN;
111  *      expm1(-INF) is -1, and
112  *      for finite argument, only expm1(0)=0 is exact.
113  *
114  * Accuracy:
115  *      according to an error analysis, the error is always less than
116  *      1 ulp (unit in the last place).
117  *
118  * Misc. info.
119  *      For IEEE double
120  *          if x >  7.09782712893383973096e+02 then expm1(x) overflow
121  *
122  * Constants:
123  * The hexadecimal values are the intended ones for the following
124  * constants. The decimal values may be used, provided that the
125  * compiler will convert from decimal to binary accurately enough
126  * to produce the hexadecimal values shown.
127  */
128 /* INDENT ON */
129
130 #include "libm_synonyms.h"      /* __expm1 */
131 #include "libm_macros.h"
132 #include <math.h>
133
134 static const double xxx[] = {
135 /* one */                1.0,
136 /* huge */               1.0e+300,
137 /* tiny */               1.0e-300,
138 /* o_threshold */        7.09782712893383973096e+02,    /* 40862E42 FEFA39EF */
139 /* ln2_hi */             6.93147180369123816490e-01,    /* 3FE62E42 FEE00000 */
140 /* ln2_lo */             1.90821492927058770002e-10,    /* 3DEA39EF 35793C76 */
141 /* invln2 */             1.44269504088896338700e+00,    /* 3FF71547 652B82FE */
142 /* scaled coefficients related to expm1 */
143 /* Q1 */                -3.33333333333331316428e-02,    /* BFA11111 111110F4 */
144 /* Q2 */                 1.58730158725481460165e-03,    /* 3F5A01A0 19FE5585 */
145 /* Q3 */                -7.93650757867487942473e-05,    /* BF14CE19 9EAADBB7 */
146 /* Q4 */                 4.00821782732936239552e-06,    /* 3ED0CFCA 86E65239 */
147 /* Q5 */                -2.01099218183624371326e-07     /* BE8AFDB7 6E09C32D */
148 };
149 #define one             xxx[0]
150 #define huge            xxx[1]
151 #define tiny            xxx[2]
152 #define o_threshold     xxx[3]
153 #define ln2_hi          xxx[4]
154 #define ln2_lo          xxx[5]
155 #define invln2          xxx[6]
156 #define Q1              xxx[7]
157 #define Q2              xxx[8]
158 #define Q3              xxx[9]
159 #define Q4              xxx[10]
160 #define Q5              xxx[11]
161
162 double
163 expm1(double x) {
164         double y, hi, lo, c = 0.0L, t, e, hxs, hfx, r1;
165         int k, xsb;
166         unsigned hx;
167
168         hx = ((unsigned *) &x)[HIWORD];             /* high word of x */
169         xsb = hx & 0x80000000;                      /* sign bit of x */
170         if (xsb == 0)
171                 y = x;
172         else
173                 y = -x;                         /* y = |x| */
174         hx &= 0x7fffffff;                   /* high word of |x| */
175
176         /* filter out huge and non-finite argument */
177         /* for example exp(38)-1 is approximately 3.1855932e+16 */
178         if (hx >= 0x4043687A) {
179                 /* if |x|>=56*ln2 (~38.8162...) */
180                 if (hx >= 0x40862E42) {              /* if |x|>=709.78... -> inf */
181                         if (hx >= 0x7ff00000) {
182                                 if (((hx & 0xfffff) | ((int *) &x)[LOWORD])
183                                         != 0)
184                                         return (x * x); /* + -> * for Cheetah */
185                                 else
186                                         /* exp(+-inf)={inf,-1} */
187                                         return (xsb == 0 ? x : -1.0);
188                         }
189                         if (x > o_threshold)
190                                 return (huge * huge);   /* overflow */
191                 }
192                 if (xsb != 0) {         /* x < -56*ln2, return -1.0 w/inexact */
193                         if (x + tiny < 0.0)          /* raise inexact */
194                                 return (tiny - one);    /* return -1 */
195                 }
196         }
197
198         /* argument reduction */
199         if (hx > 0x3fd62e42) {                       /* if  |x| > 0.5 ln2 */
200                 if (hx < 0x3FF0A2B2) {               /* and |x| < 1.5 ln2 */
201                         if (xsb == 0) {         /* positive number */
202                                 hi = x - ln2_hi;
203                                 lo = ln2_lo;
204                                 k = 1;
205                         } else {
206                                 /* negative number */
207                                 hi = x + ln2_hi;
208                                 lo = -ln2_lo;
209                                 k = -1;
210                         }
211                 } else {
212                         /* |x| > 1.5 ln2 */
213                         k = (int) (invln2 * x + (xsb == 0 ? 0.5 : -0.5));
214                         t = k;
215                         hi = x - t * ln2_hi;    /* t*ln2_hi is exact here */
216                         lo = t * ln2_lo;
217                 }
218                 x = hi - lo;
219                 c = (hi - x) - lo; /* still at |x| > 0.5 ln2 */
220         } else if (hx < 0x3c900000) {
221                 /* when |x|<2**-54, return x */
222                 t = huge + x;           /* return x w/inexact when x != 0 */
223                 return (x - (t - (huge + x)));
224         } else
225                 /* |x| <= 0.5 ln2 */
226                 k = 0;
227
228         /* x is now in primary range */
229         hfx = 0.5 * x;
230         hxs = x * hfx;
231         r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
232         t = 3.0 - r1 * hfx;
233         e = hxs * ((r1 - t) / (6.0 - x * t));
234         if (k == 0) /* |x| <= 0.5 ln2 */
235                 return (x - (x * e - hxs));
236         else {          /* |x| > 0.5 ln2 */
237                 e = (x * (e - c) - c);
238                 e -= hxs;
239                 if (k == -1)
240                         return (0.5 * (x - e) - 0.5);
241                 if (k == 1) {
242                         if (x < -0.25)
243                                 return (-2.0 * (e - (x + 0.5)));
244                         else
245                                 return (one + 2.0 * (x - e));
246                 }
247                 if (k <= -2 || k > 56) {  /* suffice to return exp(x)-1 */
248                         y = one - (e - x);
249                         ((int *) &y)[HIWORD] += k << 20;
250                         return (y - one);
251                 }
252                 t = one;
253                 if (k < 20) {
254                         ((int *) &t)[HIWORD] = 0x3ff00000 - (0x200000 >> k);
255                                                         /* t = 1 - 2^-k */
256                         y = t - (e - x);
257                         ((int *) &y)[HIWORD] += k << 20;
258                 } else {
259                         ((int *) &t)[HIWORD] = (0x3ff - k) << 20; /* 2^-k */
260                         y = x - (e + t);
261                         y += one;
262                         ((int *) &y)[HIWORD] += k << 20;
263                 }
264         }
265         return (y);
266 }
--- EOF ---

```