1/* Copyright JS Foundation and other contributors, http://js.foundation
2 *
3 * Licensed under the Apache License, Version 2.0 (the "License");
4 * you may not use this file except in compliance with the License.
5 * You may obtain a copy of the License at
6 *
7 *     http://www.apache.org/licenses/LICENSE-2.0
8 *
9 * Unless required by applicable law or agreed to in writing, software
10 * distributed under the License is distributed on an "AS IS" BASIS
11 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 * See the License for the specific language governing permissions and
13 * limitations under the License.
14 *
15 * This file is based on work under the following copyright and permission
16 * notice:
17 *
18 *     Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
19 *
20 *     Permission to use, copy, modify, and distribute this
21 *     software is freely granted, provided that this notice
22 *     is preserved.
23 *
24 *     @(#)s_expm1.c 5.1 93/09/24
25 */
26
27#include "jerry-libm-internal.h"
28
29/* expm1(x)
30 * Returns exp(x)-1, the exponential of x minus 1.
31 *
32 * Method
33 *   1. Argument reduction:
34 *  Given x, find r and integer k such that
35 *
36 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
37 *
38 *      Here a correction term c will be computed to compensate
39 *  the error in r when rounded to a floating-point number.
40 *
41 *   2. Approximating expm1(r) by a special rational function on
42 *  the interval [0,0.34658]:
43 *  Since
44 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
45 *  we define R1(r*r) by
46 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
47 *  That is,
48 *      R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
49 *         = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
50 *         = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
51 *      We use a special Reme algorithm on [0,0.347] to generate
52 *   a polynomial of degree 5 in r*r to approximate R1. The
53 *  maximum error of this polynomial approximation is bounded
54 *  by 2**-61. In other words,
55 *      R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
56 *  where   Q1  =  -1.6666666666666567384E-2,
57 *     Q2  =   3.9682539681370365873E-4,
58 *     Q3  =  -9.9206344733435987357E-6,
59 *     Q4  =   2.5051361420808517002E-7,
60 *     Q5  =  -6.2843505682382617102E-9;
61 *    z   =  r*r,
62 *  with error bounded by
63 *      |                  5           |     -61
64 *      | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
65 *      |                              |
66 *
67 *  expm1(r) = exp(r)-1 is then computed by the following
68 *   specific way which minimize the accumulation rounding error:
69 *                        2     3
70 *                        r     r    [ 3 - (R1 + R1*r/2)  ]
71 *        expm1(r) = r + --- + --- * [--------------------]
72 *                        2     2    [ 6 - r*(3 - R1*r/2) ]
73 *
74 *  To compensate the error in the argument reduction, we use
75 *    expm1(r+c) = expm1(r) + c + expm1(r)*c
76 *         ~ expm1(r) + c + r*c
77 *  Thus c+r*c will be added in as the correction terms for
78 *  expm1(r+c). Now rearrange the term to avoid optimization
79 *   screw up:
80 *                  (      2                                    2 )
81 *                  ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
82 *   expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
83 *                  ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
84 *                  (                                             )
85 *
86 *       = r - E
87 *   3. Scale back to obtain expm1(x):
88 *  From step 1, we have
89 *     expm1(x) = either 2^k*[expm1(r)+1] - 1
90 *              = or     2^k*[expm1(r) + (1-2^-k)]
91 *   4. Implementation notes:
92 *  (A). To save one multiplication, we scale the coefficient Qi
93 *       to Qi*2^i, and replace z by (x^2)/2.
94 *  (B). To achieve maximum accuracy, we compute expm1(x) by
95 *    (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
96 *    (ii)  if k=0, return r-E
97 *    (iii) if k=-1, return 0.5*(r-E)-0.5
98 *    (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
99 *                  else       return  1.0+2.0*(r-E);
100 *    (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
101 *    (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
102 *    (vii) return 2^k(1-((E+2^-k)-r))
103 *
104 * Special cases:
105 *  expm1(INF) is INF, expm1(NaN) is NaN;
106 *  expm1(-INF) is -1, and
107 *  for finite argument, only expm1(0)=0 is exact.
108 *
109 * Accuracy:
110 *  according to an error analysis, the error is always less than
111 *  1 ulp (unit in the last place).
112 *
113 * Misc. info.
114 *  For IEEE double
115 *      if x >  7.09782712893383973096e+02 then expm1(x) overflow
116 *
117 * Constants:
118 * The hexadecimal values are the intended ones for the following
119 * constants. The decimal values may be used, provided that the
120 * compiler will convert from decimal to binary accurately enough
121 * to produce the hexadecimal values shown.
122 */
123
124#define one 1.0
125#define huge 1.0e+300
126#define tiny 1.0e-300
127#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
128#define ln2_hi 6.93147180369123816490e-01      /* 0x3fe62e42, 0xfee00000 */
129#define ln2_lo 1.90821492927058770002e-10      /* 0x3dea39ef, 0x35793c76 */
130#define invln2 1.44269504088896338700e+00      /* 0x3ff71547, 0x652b82fe */
131
132/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
133#define Q1 -3.33333333333331316428e-02 /* BFA11111 111110F4 */
134#define Q2 1.58730158725481460165e-03  /* 3F5A01A0 19FE5585 */
135#define Q3 -7.93650757867487942473e-05 /* BF14CE19 9EAADBB7 */
136#define Q4 4.00821782732936239552e-06  /* 3ED0CFCA 86E65239 */
137#define Q5 -2.01099218183624371326e-07 /* BE8AFDB7 6E09C32D */
138
139double
140expm1 (double x)
141{
142  double y, hi, lo, c, e, hxs, hfx, r1;
143  double_accessor t, twopk;
144  int k, xsb;
145  unsigned int hx;
146
147  hx = __HI (x);
148  xsb = hx & 0x80000000; /* sign bit of x */
149  hx &= 0x7fffffff;      /* high word of |x| */
150
151  /* filter out huge and non-finite argument */
152  if (hx >= 0x4043687A)
153  {
154    /* if |x|>=56*ln2 */
155    if (hx >= 0x40862E42)
156    {
157      /* if |x|>=709.78... */
158      if (hx >= 0x7ff00000)
159      {
160        unsigned int low;
161        low = __LO (x);
162        if (((hx & 0xfffff) | low) != 0)
163        {
164          /* NaN */
165          return x + x;
166        }
167        else
168        {
169          /* exp(+-inf)-1={inf,-1} */
170          return (xsb == 0) ? x : -1.0;
171        }
172      }
173      if (x > o_threshold)
174      {
175        /* overflow */
176        return huge * huge;
177      }
178    }
179    if (xsb != 0)
180    {
181      /* x < -56*ln2, return -1.0 with inexact */
182      if (x + tiny < 0.0) /* raise inexact */
183      {
184        /* return -1 */
185        return tiny - one;
186      }
187    }
188  }
189
190  /* argument reduction */
191  if (hx > 0x3fd62e42)
192  {
193    /* if  |x| > 0.5 ln2 */
194    if (hx < 0x3FF0A2B2)
195    {
196      /* and |x| < 1.5 ln2 */
197      if (xsb == 0)
198      {
199        hi = x - ln2_hi;
200        lo = ln2_lo;
201        k = 1;
202      }
203      else
204      {
205        hi = x + ln2_hi;
206        lo = -ln2_lo;
207        k = -1;
208      }
209    }
210    else
211    {
212      k = (int) (invln2 * x + ((xsb == 0) ? 0.5 : -0.5));
213      t.dbl = k;
214      hi = x - t.dbl * ln2_hi; /* t*ln2_hi is exact here */
215      lo = t.dbl * ln2_lo;
216    }
217    x = hi - lo;
218    c = (hi - x) - lo;
219  }
220  else if (hx < 0x3c900000)
221  {
222    /* when |x|<2**-54, return x */
223    return x;
224  }
225  else
226  {
227    k = 0;
228  }
229
230  /* x is now in primary range */
231  hfx = 0.5 * x;
232  hxs = x * hfx;
233  r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
234  t.dbl = 3.0 - r1 * hfx;
235  e = hxs * ((r1 - t.dbl) / (6.0 - x * t.dbl));
236  if (k == 0)
237  {
238    /* c is 0 */
239    return x - (x * e - hxs);
240  }
241  else
242  {
243    twopk.as_int.hi = 0x3ff00000 + ((unsigned int) k << 20); /* 2^k */
244    twopk.as_int.lo = 0;
245    e = (x * (e - c) - c);
246    e -= hxs;
247    if (k == -1)
248    {
249      return 0.5 * (x - e) - 0.5;
250    }
251    if (k == 1)
252    {
253      if (x < -0.25)
254      {
255        return -2.0 * (e - (x + 0.5));
256      }
257      else
258      {
259        return one + 2.0 * (x - e);
260      }
261    }
262    if ((k <= -2) || (k > 56))
263    {
264      /* suffice to return exp(x)-1 */
265      y = one - (e - x);
266      if (k == 1024)
267      {
268        y = y * 2.0 * 0x1p1023;
269      }
270      else
271      {
272        y = y * twopk.dbl;
273      }
274      return y - one;
275    }
276    t.dbl = one;
277    if (k < 20)
278    {
279      t.as_int.hi = (0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
280      y = t.dbl - (e - x);
281      y = y * twopk.dbl;
282    }
283    else
284    {
285      t.as_int.hi = ((0x3ff - k) << 20); /* 2^-k */
286      y = x - (e + t.dbl);
287      y += one;
288      y = y * twopk.dbl;
289    }
290  }
291  return y;
292} /* expm1 */
293
294#undef one
295#undef huge
296#undef tiny
297#undef o_threshold
298#undef ln2_hi
299#undef ln2_lo
300#undef invln2
301#undef Q1
302#undef Q2
303#undef Q3
304#undef Q4
305#undef Q5
306