POK
expm1.c
1 /*
2  * POK header
3  *
4  * The following file is a part of the POK project. Any modification should
5  * made according to the POK licence. You CANNOT use this file or a part of
6  * this file is this part of a file for your own project
7  *
8  * For more information on the POK licence, please see our LICENCE FILE
9  *
10  * Please follow the coding guidelines described in doc/CODING_GUIDELINES
11  *
12  * Copyright (c) 2007-2009 POK team
13  *
14  * Created by julien on Fri Jan 30 14:41:34 2009
15  */
16 
17 /* @(#)s_expm1.c 5.1 93/09/24 */
18 /*
19  * ====================================================
20  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
21  *
22  * Developed at SunPro, a Sun Microsystems, Inc. business.
23  * Permission to use, copy, modify, and distribute this
24  * software is freely granted, provided that this notice
25  * is preserved.
26  * ====================================================
27  */
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  * (where z=r*r, and the values of Q1 to Q5 are listed below)
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 #ifdef POK_NEEDS_LIBMATH
125 
126 #include <libm.h>
127 #include "math_private.h"
128 
129 static const double
130 one = 1.0,
131 huge = 1.0e+300,
132 tiny = 1.0e-300,
133 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
134 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
135 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
136 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
137  /* scaled coefficients related to expm1 */
138 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
139 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
140 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
141 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
142 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
143 
144 double
145 expm1(double x)
146 {
147  double y,hi,lo,c,t,e,hxs,hfx,r1;
148  int32_t k,xsb;
149  uint32_t hx;
150 
151  c = 0;
152  GET_HIGH_WORD(hx,x);
153  xsb = hx&0x80000000; /* sign bit of x */
154  if(xsb==0) y=x; else y= -x; /* y = |x| */
155  hx &= 0x7fffffff; /* high word of |x| */
156 
157  /* filter out huge and non-finite argument */
158  if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
159  if(hx >= 0x40862E42) { /* if |x|>=709.78... */
160  if(hx>=0x7ff00000) {
161  uint32_t low;
162  GET_LOW_WORD(low,x);
163  if(((hx&0xfffff)|low)!=0)
164  return x+x; /* NaN */
165  else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
166  }
167  if(x > o_threshold) return huge*huge; /* overflow */
168  }
169  if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
170  if(x+tiny<0.0) /* raise inexact */
171  return tiny-one; /* return -1 */
172  }
173  }
174 
175  /* argument reduction */
176  if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
177  if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
178  if(xsb==0)
179  {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
180  else
181  {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
182  } else {
183  k = invln2*x+((xsb==0)?0.5:-0.5);
184  t = k;
185  hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
186  lo = t*ln2_lo;
187  }
188  x = hi - lo;
189  c = (hi-x)-lo;
190  }
191  else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
192  t = huge+x; /* return x with inexact flags when x!=0 */
193  return x - (t-(huge+x));
194  }
195  else k = 0;
196 
197  /* x is now in primary range */
198  hfx = 0.5*x;
199  hxs = x*hfx;
200  r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
201  t = 3.0-r1*hfx;
202  e = hxs*((r1-t)/(6.0 - x*t));
203  if(k==0) return x - (x*e-hxs); /* c is 0 */
204  else {
205  e = (x*(e-c)-c);
206  e -= hxs;
207  if(k== -1) return 0.5*(x-e)-0.5;
208  if(k==1) {
209  if(x < -0.25) return -2.0*(e-(x+0.5));
210  else return one+2.0*(x-e);
211  }
212  if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
213  uint32_t high;
214  y = one-(e-x);
215  GET_HIGH_WORD(high,y);
216  SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
217  return y-one;
218  }
219  t = one;
220  if(k<20) {
221  uint32_t high;
222  SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
223  y = t-(e-x);
224  GET_HIGH_WORD(high,y);
225  SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
226  } else {
227  uint32_t high;
228  SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
229  y = x-(e+t);
230  y += one;
231  GET_HIGH_WORD(high,y);
232  SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
233  }
234  }
235  return y;
236 }
237 
238 #endif