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2aa62f2b | 1 | /* @(#)s_expm1.c 5.1 93/09/24 */\r |
2 | /*\r | |
3 | * ====================================================\r | |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\r | |
5 | *\r | |
6 | * Developed at SunPro, a Sun Microsystems, Inc. business.\r | |
7 | * Permission to use, copy, modify, and distribute this\r | |
8 | * software is freely granted, provided that this notice\r | |
9 | * is preserved.\r | |
10 | * ====================================================\r | |
11 | */\r | |
12 | #include <LibConfig.h>\r | |
13 | #include <sys/EfiCdefs.h>\r | |
14 | #if defined(LIBM_SCCS) && !defined(lint)\r | |
15 | __RCSID("$NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $");\r | |
16 | #endif\r | |
17 | \r | |
18 | #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */\r | |
19 | // C4756: overflow in constant arithmetic\r | |
20 | #pragma warning ( disable : 4756 )\r | |
21 | #endif\r | |
22 | \r | |
23 | /* expm1(x)\r | |
24 | * Returns exp(x)-1, the exponential of x minus 1.\r | |
25 | *\r | |
26 | * Method\r | |
27 | * 1. Argument reduction:\r | |
28 | * Given x, find r and integer k such that\r | |
29 | *\r | |
30 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658\r | |
31 | *\r | |
32 | * Here a correction term c will be computed to compensate\r | |
33 | * the error in r when rounded to a floating-point number.\r | |
34 | *\r | |
35 | * 2. Approximating expm1(r) by a special rational function on\r | |
36 | * the interval [0,0.34658]:\r | |
37 | * Since\r | |
38 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...\r | |
39 | * we define R1(r*r) by\r | |
40 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)\r | |
41 | * That is,\r | |
42 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)\r | |
43 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))\r | |
44 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...\r | |
45 | * We use a special Reme algorithm on [0,0.347] to generate\r | |
46 | * a polynomial of degree 5 in r*r to approximate R1. The\r | |
47 | * maximum error of this polynomial approximation is bounded\r | |
48 | * by 2**-61. In other words,\r | |
49 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5\r | |
50 | * where Q1 = -1.6666666666666567384E-2,\r | |
51 | * Q2 = 3.9682539681370365873E-4,\r | |
52 | * Q3 = -9.9206344733435987357E-6,\r | |
53 | * Q4 = 2.5051361420808517002E-7,\r | |
54 | * Q5 = -6.2843505682382617102E-9;\r | |
55 | * (where z=r*r, and the values of Q1 to Q5 are listed below)\r | |
56 | * with error bounded by\r | |
57 | * | 5 | -61\r | |
58 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2\r | |
59 | * | |\r | |
60 | *\r | |
61 | * expm1(r) = exp(r)-1 is then computed by the following\r | |
62 | * specific way which minimize the accumulation rounding error:\r | |
63 | * 2 3\r | |
64 | * r r [ 3 - (R1 + R1*r/2) ]\r | |
65 | * expm1(r) = r + --- + --- * [--------------------]\r | |
66 | * 2 2 [ 6 - r*(3 - R1*r/2) ]\r | |
67 | *\r | |
68 | * To compensate the error in the argument reduction, we use\r | |
69 | * expm1(r+c) = expm1(r) + c + expm1(r)*c\r | |
70 | * ~ expm1(r) + c + r*c\r | |
71 | * Thus c+r*c will be added in as the correction terms for\r | |
72 | * expm1(r+c). Now rearrange the term to avoid optimization\r | |
73 | * screw up:\r | |
74 | * ( 2 2 )\r | |
75 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )\r | |
76 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )\r | |
77 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )\r | |
78 | * ( )\r | |
79 | *\r | |
80 | * = r - E\r | |
81 | * 3. Scale back to obtain expm1(x):\r | |
82 | * From step 1, we have\r | |
83 | * expm1(x) = either 2^k*[expm1(r)+1] - 1\r | |
84 | * = or 2^k*[expm1(r) + (1-2^-k)]\r | |
85 | * 4. Implementation notes:\r | |
86 | * (A). To save one multiplication, we scale the coefficient Qi\r | |
87 | * to Qi*2^i, and replace z by (x^2)/2.\r | |
88 | * (B). To achieve maximum accuracy, we compute expm1(x) by\r | |
89 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)\r | |
90 | * (ii) if k=0, return r-E\r | |
91 | * (iii) if k=-1, return 0.5*(r-E)-0.5\r | |
92 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)\r | |
93 | * else return 1.0+2.0*(r-E);\r | |
94 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)\r | |
95 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else\r | |
96 | * (vii) return 2^k(1-((E+2^-k)-r))\r | |
97 | *\r | |
98 | * Special cases:\r | |
99 | * expm1(INF) is INF, expm1(NaN) is NaN;\r | |
100 | * expm1(-INF) is -1, and\r | |
101 | * for finite argument, only expm1(0)=0 is exact.\r | |
102 | *\r | |
103 | * Accuracy:\r | |
104 | * according to an error analysis, the error is always less than\r | |
105 | * 1 ulp (unit in the last place).\r | |
106 | *\r | |
107 | * Misc. info.\r | |
108 | * For IEEE double\r | |
109 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow\r | |
110 | *\r | |
111 | * Constants:\r | |
112 | * The hexadecimal values are the intended ones for the following\r | |
113 | * constants. The decimal values may be used, provided that the\r | |
114 | * compiler will convert from decimal to binary accurately enough\r | |
115 | * to produce the hexadecimal values shown.\r | |
116 | */\r | |
117 | \r | |
118 | #include "math.h"\r | |
119 | #include "math_private.h"\r | |
120 | \r | |
121 | static const double\r | |
122 | one = 1.0,\r | |
123 | huge = 1.0e+300,\r | |
124 | tiny = 1.0e-300,\r | |
125 | o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */\r | |
126 | ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */\r | |
127 | ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */\r | |
128 | invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */\r | |
129 | /* scaled coefficients related to expm1 */\r | |
130 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */\r | |
131 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */\r | |
132 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */\r | |
133 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */\r | |
134 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */\r | |
135 | \r | |
136 | double\r | |
137 | expm1(double x)\r | |
138 | {\r | |
139 | double y,hi,lo,c,t,e,hxs,hfx,r1;\r | |
140 | int32_t k,xsb;\r | |
141 | u_int32_t hx;\r | |
142 | \r | |
143 | c = 0;\r | |
144 | GET_HIGH_WORD(hx,x);\r | |
145 | xsb = hx&0x80000000; /* sign bit of x */\r | |
146 | if(xsb==0) y=x; else y= -x; /* y = |x| */\r | |
147 | hx &= 0x7fffffff; /* high word of |x| */\r | |
148 | \r | |
149 | /* filter out huge and non-finite argument */\r | |
150 | if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */\r | |
151 | if(hx >= 0x40862E42) { /* if |x|>=709.78... */\r | |
152 | if(hx>=0x7ff00000) {\r | |
153 | u_int32_t low;\r | |
154 | GET_LOW_WORD(low,x);\r | |
155 | if(((hx&0xfffff)|low)!=0)\r | |
156 | return x+x; /* NaN */\r | |
157 | else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */\r | |
158 | }\r | |
159 | if(x > o_threshold) return huge*huge; /* overflow */\r | |
160 | }\r | |
161 | if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */\r | |
162 | if(x+tiny<0.0) /* raise inexact */\r | |
163 | return tiny-one; /* return -1 */\r | |
164 | }\r | |
165 | }\r | |
166 | \r | |
167 | /* argument reduction */\r | |
168 | if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */\r | |
169 | if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */\r | |
170 | if(xsb==0)\r | |
171 | {hi = x - ln2_hi; lo = ln2_lo; k = 1;}\r | |
172 | else\r | |
173 | {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}\r | |
174 | } else {\r | |
175 | k = (int32_t)(invln2*x+((xsb==0)?0.5:-0.5));\r | |
176 | t = k;\r | |
177 | hi = x - t*ln2_hi; /* t*ln2_hi is exact here */\r | |
178 | lo = t*ln2_lo;\r | |
179 | }\r | |
180 | x = hi - lo;\r | |
181 | c = (hi-x)-lo;\r | |
182 | }\r | |
183 | else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */\r | |
184 | t = huge+x; /* return x with inexact flags when x!=0 */\r | |
185 | return x - (t-(huge+x));\r | |
186 | }\r | |
187 | else k = 0;\r | |
188 | \r | |
189 | /* x is now in primary range */\r | |
190 | hfx = 0.5*x;\r | |
191 | hxs = x*hfx;\r | |
192 | r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));\r | |
193 | t = 3.0-r1*hfx;\r | |
194 | e = hxs*((r1-t)/(6.0 - x*t));\r | |
195 | if(k==0) return x - (x*e-hxs); /* c is 0 */\r | |
196 | else {\r | |
197 | e = (x*(e-c)-c);\r | |
198 | e -= hxs;\r | |
199 | if(k== -1) return 0.5*(x-e)-0.5;\r | |
200 | if(k==1) {\r | |
201 | if(x < -0.25) return -2.0*(e-(x+0.5));\r | |
202 | else return one+2.0*(x-e);\r | |
203 | }\r | |
204 | if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */\r | |
205 | u_int32_t high;\r | |
206 | y = one-(e-x);\r | |
207 | GET_HIGH_WORD(high,y);\r | |
208 | SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */\r | |
209 | return y-one;\r | |
210 | }\r | |
211 | t = one;\r | |
212 | if(k<20) {\r | |
213 | u_int32_t high;\r | |
214 | SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */\r | |
215 | y = t-(e-x);\r | |
216 | GET_HIGH_WORD(high,y);\r | |
217 | SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */\r | |
218 | } else {\r | |
219 | u_int32_t high;\r | |
220 | SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */\r | |
221 | y = x-(e+t);\r | |
222 | y += one;\r | |
223 | GET_HIGH_WORD(high,y);\r | |
224 | SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */\r | |
225 | }\r | |
226 | }\r | |
227 | return y;\r | |
228 | }\r |