1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /* expm1(x)
13 * Returns exp(x)-1, the exponential of x minus 1.
14 *
15 * Method
16 * 1. Argument reduction:
17 * Given x, find r and integer k such that
18 *
19 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
20 *
21 * Here a correction term c will be computed to compensate
22 * the error in r when rounded to a floating-point number.
23 *
24 * 2. Approximating expm1(r) by a special rational function on
25 * the interval [0,0.34658]:
26 * Since
27 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
28 * we define R1(r*r) by
29 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
30 * That is,
31 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
32 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
33 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
34 * We use a special Reme algorithm on [0,0.347] to generate
35 * a polynomial of degree 5 in r*r to approximate R1. The
36 * maximum error of this polynomial approximation is bounded
37 * by 2**-61. In other words,
38 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
39 * where Q1 = -1.6666666666666567384E-2,
40 * Q2 = 3.9682539681370365873E-4,
41 * Q3 = -9.9206344733435987357E-6,
42 * Q4 = 2.5051361420808517002E-7,
43 * Q5 = -6.2843505682382617102E-9;
44 * z = r*r,
45 * with error bounded by
46 * | 5 | -61
47 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
48 * | |
49 *
50 * expm1(r) = exp(r)-1 is then computed by the following
51 * specific way which minimize the accumulation rounding error:
52 * 2 3
53 * r r [ 3 - (R1 + R1*r/2) ]
54 * expm1(r) = r + --- + --- * [--------------------]
55 * 2 2 [ 6 - r*(3 - R1*r/2) ]
56 *
57 * To compensate the error in the argument reduction, we use
58 * expm1(r+c) = expm1(r) + c + expm1(r)*c
59 * ~ expm1(r) + c + r*c
60 * Thus c+r*c will be added in as the correction terms for
61 * expm1(r+c). Now rearrange the term to avoid optimization
62 * screw up:
63 * ( 2 2 )
64 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
65 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
66 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
67 * ( )
68 *
69 * = r - E
70 * 3. Scale back to obtain expm1(x):
71 * From step 1, we have
72 * expm1(x) = either 2^k*[expm1(r)+1] - 1
73 * = or 2^k*[expm1(r) + (1-2^-k)]
74 * 4. Implementation notes:
75 * (A). To save one multiplication, we scale the coefficient Qi
76 * to Qi*2^i, and replace z by (x^2)/2.
77 * (B). To achieve maximum accuracy, we compute expm1(x) by
78 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
79 * (ii) if k=0, return r-E
80 * (iii) if k=-1, return 0.5*(r-E)-0.5
81 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
82 * else return 1.0+2.0*(r-E);
83 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
84 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
85 * (vii) return 2^k(1-((E+2^-k)-r))
86 *
87 * Special cases:
88 * expm1(INF) is INF, expm1(NaN) is NaN;
89 * expm1(-INF) is -1, and
90 * for finite argument, only expm1(0)=0 is exact.
91 *
92 * Accuracy:
93 * according to an error analysis, the error is always less than
94 * 1 ulp (unit in the last place).
95 *
96 * Misc. info.
97 * For IEEE double
98 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
99 *
100 * Constants:
101 * The hexadecimal values are the intended ones for the following
102 * constants. The decimal values may be used, provided that the
103 * compiler will convert from decimal to binary accurately enough
104 * to produce the hexadecimal values shown.
105 */
106
107 #include <float.h>
108
109 #include "math.h"
110 #include "math_private.h"
111
112 static const double
113 one = 1.0,
114 tiny = 1.0e-300,
115 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
116 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
117 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
118 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
119 /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
120 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
121 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
122 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
123 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
124 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
125
126 static volatile double huge = 1.0e+300;
127
128 double
expm1(double x)129 expm1(double x)
130 {
131 double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
132 int32_t k,xsb;
133 u_int32_t hx;
134
135 GET_HIGH_WORD(hx,x);
136 xsb = hx&0x80000000; /* sign bit of x */
137 hx &= 0x7fffffff; /* high word of |x| */
138
139 /* filter out huge and non-finite argument */
140 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
141 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
142 if(hx>=0x7ff00000) {
143 u_int32_t low;
144 GET_LOW_WORD(low,x);
145 if(((hx&0xfffff)|low)!=0)
146 return x+x; /* NaN */
147 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
148 }
149 if(x > o_threshold) return huge*huge; /* overflow */
150 }
151 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
152 if(x+tiny<0.0) /* raise inexact */
153 return tiny-one; /* return -1 */
154 }
155 }
156
157 /* argument reduction */
158 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
159 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
160 if(xsb==0)
161 {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
162 else
163 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
164 } else {
165 k = invln2*x+((xsb==0)?0.5:-0.5);
166 t = k;
167 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
168 lo = t*ln2_lo;
169 }
170 STRICT_ASSIGN(double, x, hi - lo);
171 c = (hi-x)-lo;
172 }
173 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
174 t = huge+x; /* return x with inexact flags when x!=0 */
175 return x - (t-(huge+x));
176 }
177 else k = 0;
178
179 /* x is now in primary range */
180 hfx = 0.5*x;
181 hxs = x*hfx;
182 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
183 t = 3.0-r1*hfx;
184 e = hxs*((r1-t)/(6.0 - x*t));
185 if(k==0) return x - (x*e-hxs); /* c is 0 */
186 else {
187 INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */
188 e = (x*(e-c)-c);
189 e -= hxs;
190 if(k== -1) return 0.5*(x-e)-0.5;
191 if(k==1) {
192 if(x < -0.25) return -2.0*(e-(x+0.5));
193 else return one+2.0*(x-e);
194 }
195 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
196 y = one-(e-x);
197 if (k == 1024) y = y*2.0*0x1p1023;
198 else y = y*twopk;
199 return y-one;
200 }
201 t = one;
202 if(k<20) {
203 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
204 y = t-(e-x);
205 y = y*twopk;
206 } else {
207 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
208 y = x-(e+t);
209 y += one;
210 y = y*twopk;
211 }
212 }
213 return y;
214 }
215
216 #if (LDBL_MANT_DIG == 53)
217 __weak_reference(expm1, expm1l);
218 #endif
219