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2aa62f2b | 1 | /* @(#)e_exp.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: e_exp.c,v 1.11 2002/05/26 22:01:49 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 | /* __ieee754_exp(x)\r | |
24 | * Returns the exponential of x.\r | |
25 | *\r | |
26 | * Method\r | |
27 | * 1. Argument reduction:\r | |
28 | * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.\r | |
29 | * Given x, find r and integer k such that\r | |
30 | *\r | |
31 | * x = k*ln2 + r, |r| <= 0.5*ln2.\r | |
32 | *\r | |
33 | * Here r will be represented as r = hi-lo for better\r | |
34 | * accuracy.\r | |
35 | *\r | |
36 | * 2. Approximation of exp(r) by a special rational function on\r | |
37 | * the interval [0,0.34658]:\r | |
38 | * Write\r | |
39 | * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...\r | |
40 | * We use a special Reme algorithm on [0,0.34658] to generate\r | |
41 | * a polynomial of degree 5 to approximate R. The maximum error\r | |
42 | * of this polynomial approximation is bounded by 2**-59. In\r | |
43 | * other words,\r | |
44 | * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5\r | |
45 | * (where z=r*r, and the values of P1 to P5 are listed below)\r | |
46 | * and\r | |
47 | * | 5 | -59\r | |
48 | * | 2.0+P1*z+...+P5*z - R(z) | <= 2\r | |
49 | * | |\r | |
50 | * The computation of exp(r) thus becomes\r | |
51 | * 2*r\r | |
52 | * exp(r) = 1 + -------\r | |
53 | * R - r\r | |
54 | * r*R1(r)\r | |
55 | * = 1 + r + ----------- (for better accuracy)\r | |
56 | * 2 - R1(r)\r | |
57 | * where\r | |
58 | * 2 4 10\r | |
59 | * R1(r) = r - (P1*r + P2*r + ... + P5*r ).\r | |
60 | *\r | |
61 | * 3. Scale back to obtain exp(x):\r | |
62 | * From step 1, we have\r | |
63 | * exp(x) = 2^k * exp(r)\r | |
64 | *\r | |
65 | * Special cases:\r | |
66 | * exp(INF) is INF, exp(NaN) is NaN;\r | |
67 | * exp(-INF) is 0, and\r | |
68 | * for finite argument, only exp(0)=1 is exact.\r | |
69 | *\r | |
70 | * Accuracy:\r | |
71 | * according to an error analysis, the error is always less than\r | |
72 | * 1 ulp (unit in the last place).\r | |
73 | *\r | |
74 | * Misc. info.\r | |
75 | * For IEEE double\r | |
76 | * if x > 7.09782712893383973096e+02 then exp(x) overflow\r | |
77 | * if x < -7.45133219101941108420e+02 then exp(x) underflow\r | |
78 | *\r | |
79 | * Constants:\r | |
80 | * The hexadecimal values are the intended ones for the following\r | |
81 | * constants. The decimal values may be used, provided that the\r | |
82 | * compiler will convert from decimal to binary accurately enough\r | |
83 | * to produce the hexadecimal values shown.\r | |
84 | */\r | |
85 | \r | |
86 | #include "math.h"\r | |
87 | #include "math_private.h"\r | |
88 | \r | |
89 | static const double\r | |
90 | one = 1.0,\r | |
91 | halF[2] = {0.5,-0.5,},\r | |
92 | huge = 1.0e+300,\r | |
93 | twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/\r | |
94 | o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */\r | |
95 | u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */\r | |
96 | ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */\r | |
97 | -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */\r | |
98 | ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */\r | |
99 | -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */\r | |
100 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */\r | |
101 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */\r | |
102 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */\r | |
103 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */\r | |
104 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */\r | |
105 | P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */\r | |
106 | \r | |
107 | \r | |
108 | double\r | |
109 | __ieee754_exp(double x) /* default IEEE double exp */\r | |
110 | {\r | |
111 | double y,hi,lo,c,t;\r | |
112 | int32_t k,xsb;\r | |
113 | u_int32_t hx;\r | |
114 | \r | |
115 | hi = lo = 0;\r | |
116 | k = 0;\r | |
117 | GET_HIGH_WORD(hx,x);\r | |
118 | xsb = (hx>>31)&1; /* sign bit of x */\r | |
119 | hx &= 0x7fffffff; /* high word of |x| */\r | |
120 | \r | |
121 | /* filter out non-finite argument */\r | |
122 | if(hx >= 0x40862E42) { /* if |x|>=709.78... */\r | |
123 | if(hx>=0x7ff00000) {\r | |
124 | u_int32_t lx;\r | |
125 | GET_LOW_WORD(lx,x);\r | |
126 | if(((hx&0xfffff)|lx)!=0)\r | |
127 | return x+x; /* NaN */\r | |
128 | else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */\r | |
129 | }\r | |
130 | if(x > o_threshold) return huge*huge; /* overflow */\r | |
131 | if(x < u_threshold) return twom1000*twom1000; /* underflow */\r | |
132 | }\r | |
133 | \r | |
134 | /* argument reduction */\r | |
135 | if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */\r | |
136 | if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */\r | |
137 | hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;\r | |
138 | } else {\r | |
139 | k = (int32_t)(invln2*x+halF[xsb]);\r | |
140 | t = k;\r | |
141 | hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */\r | |
142 | lo = t*ln2LO[0];\r | |
143 | }\r | |
144 | x = hi - lo;\r | |
145 | }\r | |
146 | else if(hx < 0x3e300000) { /* when |x|<2**-28 */\r | |
147 | if(huge+x>one) return one+x;/* trigger inexact */\r | |
148 | }\r | |
149 | else k = 0;\r | |
150 | \r | |
151 | /* x is now in primary range */\r | |
152 | t = x*x;\r | |
153 | c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));\r | |
154 | if(k==0) return one-((x*c)/(c-2.0)-x);\r | |
155 | else y = one-((lo-(x*c)/(2.0-c))-hi);\r | |
156 | if(k >= -1021) {\r | |
157 | u_int32_t hy;\r | |
158 | GET_HIGH_WORD(hy,y);\r | |
159 | SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */\r | |
160 | return y;\r | |
161 | } else {\r | |
162 | u_int32_t hy;\r | |
163 | GET_HIGH_WORD(hy,y);\r | |
164 | SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */\r | |
165 | return y*twom1000;\r | |
166 | }\r | |
167 | }\r |