]> git.proxmox.com Git - mirror_edk2.git/blob - StdLib/LibC/Math/s_expm1.c
ArmPkg: ArmLib: purge incorrect ArmDrainWriteBuffer () alias
[mirror_edk2.git] / StdLib / LibC / Math / s_expm1.c
1 /* @(#)s_expm1.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12 #include <LibConfig.h>
13 #include <sys/EfiCdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $");
16 #endif
17
18 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
19 // C4756: overflow in constant arithmetic
20 #pragma warning ( disable : 4756 )
21 #endif
22
23 /* expm1(x)
24 * Returns exp(x)-1, the exponential of x minus 1.
25 *
26 * Method
27 * 1. Argument reduction:
28 * Given x, find r and integer k such that
29 *
30 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
31 *
32 * Here a correction term c will be computed to compensate
33 * the error in r when rounded to a floating-point number.
34 *
35 * 2. Approximating expm1(r) by a special rational function on
36 * the interval [0,0.34658]:
37 * Since
38 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
39 * we define R1(r*r) by
40 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
41 * That is,
42 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
43 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
44 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
45 * We use a special Reme algorithm on [0,0.347] to generate
46 * a polynomial of degree 5 in r*r to approximate R1. The
47 * maximum error of this polynomial approximation is bounded
48 * by 2**-61. In other words,
49 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
50 * where Q1 = -1.6666666666666567384E-2,
51 * Q2 = 3.9682539681370365873E-4,
52 * Q3 = -9.9206344733435987357E-6,
53 * Q4 = 2.5051361420808517002E-7,
54 * Q5 = -6.2843505682382617102E-9;
55 * (where z=r*r, and the values of Q1 to Q5 are listed below)
56 * with error bounded by
57 * | 5 | -61
58 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
59 * | |
60 *
61 * expm1(r) = exp(r)-1 is then computed by the following
62 * specific way which minimize the accumulation rounding error:
63 * 2 3
64 * r r [ 3 - (R1 + R1*r/2) ]
65 * expm1(r) = r + --- + --- * [--------------------]
66 * 2 2 [ 6 - r*(3 - R1*r/2) ]
67 *
68 * To compensate the error in the argument reduction, we use
69 * expm1(r+c) = expm1(r) + c + expm1(r)*c
70 * ~ expm1(r) + c + r*c
71 * Thus c+r*c will be added in as the correction terms for
72 * expm1(r+c). Now rearrange the term to avoid optimization
73 * screw up:
74 * ( 2 2 )
75 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
76 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
77 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
78 * ( )
79 *
80 * = r - E
81 * 3. Scale back to obtain expm1(x):
82 * From step 1, we have
83 * expm1(x) = either 2^k*[expm1(r)+1] - 1
84 * = or 2^k*[expm1(r) + (1-2^-k)]
85 * 4. Implementation notes:
86 * (A). To save one multiplication, we scale the coefficient Qi
87 * to Qi*2^i, and replace z by (x^2)/2.
88 * (B). To achieve maximum accuracy, we compute expm1(x) by
89 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
90 * (ii) if k=0, return r-E
91 * (iii) if k=-1, return 0.5*(r-E)-0.5
92 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
93 * else return 1.0+2.0*(r-E);
94 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
95 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
96 * (vii) return 2^k(1-((E+2^-k)-r))
97 *
98 * Special cases:
99 * expm1(INF) is INF, expm1(NaN) is NaN;
100 * expm1(-INF) is -1, and
101 * for finite argument, only expm1(0)=0 is exact.
102 *
103 * Accuracy:
104 * according to an error analysis, the error is always less than
105 * 1 ulp (unit in the last place).
106 *
107 * Misc. info.
108 * For IEEE double
109 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
110 *
111 * Constants:
112 * The hexadecimal values are the intended ones for the following
113 * constants. The decimal values may be used, provided that the
114 * compiler will convert from decimal to binary accurately enough
115 * to produce the hexadecimal values shown.
116 */
117
118 #include "math.h"
119 #include "math_private.h"
120
121 static const double
122 one = 1.0,
123 huge = 1.0e+300,
124 tiny = 1.0e-300,
125 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
126 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
127 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
128 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
129 /* scaled coefficients related to expm1 */
130 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
131 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
132 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
133 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
134 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
135
136 double
137 expm1(double x)
138 {
139 double y,hi,lo,c,t,e,hxs,hfx,r1;
140 int32_t k,xsb;
141 u_int32_t hx;
142
143 c = 0;
144 GET_HIGH_WORD(hx,x);
145 xsb = hx&0x80000000; /* sign bit of x */
146 if(xsb==0) y=x; else y= -x; /* y = |x| */
147 hx &= 0x7fffffff; /* high word of |x| */
148
149 /* filter out huge and non-finite argument */
150 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
151 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
152 if(hx>=0x7ff00000) {
153 u_int32_t low;
154 GET_LOW_WORD(low,x);
155 if(((hx&0xfffff)|low)!=0)
156 return x+x; /* NaN */
157 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
158 }
159 if(x > o_threshold) return huge*huge; /* overflow */
160 }
161 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
162 if(x+tiny<0.0) /* raise inexact */
163 return tiny-one; /* return -1 */
164 }
165 }
166
167 /* argument reduction */
168 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
169 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
170 if(xsb==0)
171 {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
172 else
173 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
174 } else {
175 k = (int32_t)(invln2*x+((xsb==0)?0.5:-0.5));
176 t = k;
177 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
178 lo = t*ln2_lo;
179 }
180 x = hi - lo;
181 c = (hi-x)-lo;
182 }
183 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
184 t = huge+x; /* return x with inexact flags when x!=0 */
185 return x - (t-(huge+x));
186 }
187 else k = 0;
188
189 /* x is now in primary range */
190 hfx = 0.5*x;
191 hxs = x*hfx;
192 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
193 t = 3.0-r1*hfx;
194 e = hxs*((r1-t)/(6.0 - x*t));
195 if(k==0) return x - (x*e-hxs); /* c is 0 */
196 else {
197 e = (x*(e-c)-c);
198 e -= hxs;
199 if(k== -1) return 0.5*(x-e)-0.5;
200 if(k==1) {
201 if(x < -0.25) return -2.0*(e-(x+0.5));
202 else return one+2.0*(x-e);
203 }
204 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
205 u_int32_t high;
206 y = one-(e-x);
207 GET_HIGH_WORD(high,y);
208 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
209 return y-one;
210 }
211 t = one;
212 if(k<20) {
213 u_int32_t high;
214 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
215 y = t-(e-x);
216 GET_HIGH_WORD(high,y);
217 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
218 } else {
219 u_int32_t high;
220 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
221 y = x-(e+t);
222 y += one;
223 GET_HIGH_WORD(high,y);
224 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
225 }
226 }
227 return y;
228 }