1
2 /* @(#)s_expm1.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14 /*
15 FUNCTION
16 <<expm1>>, <<expm1f>>---exponential minus 1
17 INDEX
18 expm1
19 INDEX
20 expm1f
21
22 SYNOPSIS
23 #include <math.h>
24 double expm1(double <[x]>);
25 float expm1f(float <[x]>);
26
27 DESCRIPTION
28 <<expm1>> and <<expm1f>> calculate the exponential of <[x]>
29 and subtract 1, that is,
30 @ifnottex
31 e raised to the power <[x]> minus 1 (where e
32 @end ifnottex
33 @tex
34 $e^x - 1$ (where $e$
35 @end tex
36 is the base of the natural system of logarithms, approximately
37 2.71828). The result is accurate even for small values of
38 <[x]>, where using <<exp(<[x]>)-1>> would lose many
39 significant digits.
40
41 RETURNS
42 e raised to the power <[x]>, minus 1.
43
44 PORTABILITY
45 Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by
46 the System V Interface Definition (Issue 2).
47 */
48
49 /* expm1(x)
50 * Returns exp(x)-1, the exponential of x minus 1.
51 *
52 * Method
53 * 1. Argument reduction:
54 * Given x, find r and integer k such that
55 *
56 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
57 *
58 * Here a correction term c will be computed to compensate
59 * the error in r when rounded to a floating-point number.
60 *
61 * 2. Approximating expm1(r) by a special rational function on
62 * the interval [0,0.34658]:
63 * Since
64 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
65 * we define R1(r*r) by
66 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
67 * That is,
68 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
69 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
70 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
71 * We use a special Remez algorithm on [0,0.347] to generate
72 * a polynomial of degree 5 in r*r to approximate R1. The
73 * maximum error of this polynomial approximation is bounded
74 * by 2**-61. In other words,
75 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
76 * where Q1 = -1.6666666666666567384E-2,
77 * Q2 = 3.9682539681370365873E-4,
78 * Q3 = -9.9206344733435987357E-6,
79 * Q4 = 2.5051361420808517002E-7,
80 * Q5 = -6.2843505682382617102E-9;
81 * (where z=r*r, and the values of Q1 to Q5 are listed below)
82 * with error bounded by
83 * | 5 | -61
84 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
85 * | |
86 *
87 * expm1(r) = exp(r)-1 is then computed by the following
88 * specific way which minimize the accumulation rounding error:
89 * 2 3
90 * r r [ 3 - (R1 + R1*r/2) ]
91 * expm1(r) = r + --- + --- * [--------------------]
92 * 2 2 [ 6 - r*(3 - R1*r/2) ]
93 *
94 * To compensate the error in the argument reduction, we use
95 * expm1(r+c) = expm1(r) + c + expm1(r)*c
96 * ~ expm1(r) + c + r*c
97 * Thus c+r*c will be added in as the correction terms for
98 * expm1(r+c). Now rearrange the term to avoid optimization
99 * screw up:
100 * ( 2 2 )
101 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
102 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
103 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
104 * ( )
105 *
106 * = r - E
107 * 3. Scale back to obtain expm1(x):
108 * From step 1, we have
109 * expm1(x) = either 2^k*[expm1(r)+1] - 1
110 * = or 2^k*[expm1(r) + (1-2^-k)]
111 * 4. Implementation notes:
112 * (A). To save one multiplication, we scale the coefficient Qi
113 * to Qi*2^i, and replace z by (x^2)/2.
114 * (B). To achieve maximum accuracy, we compute expm1(x) by
115 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
116 * (ii) if k=0, return r-E
117 * (iii) if k=-1, return 0.5*(r-E)-0.5
118 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
119 * else return 1.0+2.0*(r-E);
120 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
121 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
122 * (vii) return 2^k(1-((E+2^-k)-r))
123 *
124 * Special cases:
125 * expm1(INF) is INF, expm1(NaN) is NaN;
126 * expm1(-INF) is -1, and
127 * for finite argument, only expm1(0)=0 is exact.
128 *
129 * Accuracy:
130 * according to an error analysis, the error is always less than
131 * 1 ulp (unit in the last place).
132 *
133 * Misc. info.
134 * For IEEE double
135 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
136 *
137 * Constants:
138 * The hexadecimal values are the intended ones for the following
139 * constants. The decimal values may be used, provided that the
140 * compiler will convert from decimal to binary accurately enough
141 * to produce the hexadecimal values shown.
142 */
143
144 #include "fdlibm.h"
145 #include "math_config.h"
146
147 #ifdef _NEED_FLOAT64
148
149 static const __float64
150 one = _F_64(1.0),
151 huge = _F_64(1.0e+300),
152 tiny = _F_64(1.0e-300),
153 o_threshold = _F_64(7.09782712893383973096e+02),/* 0x40862E42, 0xFEFA39EF */
154 ln2_hi = _F_64(6.93147180369123816490e-01),/* 0x3fe62e42, 0xfee00000 */
155 ln2_lo = _F_64(1.90821492927058770002e-10),/* 0x3dea39ef, 0x35793c76 */
156 invln2 = _F_64(1.44269504088896338700e+00),/* 0x3ff71547, 0x652b82fe */
157 /* scaled coefficients related to expm1 */
158 Q1 = _F_64(-3.33333333333331316428e-02), /* BFA11111 111110F4 */
159 Q2 = _F_64(1.58730158725481460165e-03), /* 3F5A01A0 19FE5585 */
160 Q3 = _F_64(-7.93650757867487942473e-05), /* BF14CE19 9EAADBB7 */
161 Q4 = _F_64(4.00821782732936239552e-06), /* 3ED0CFCA 86E65239 */
162 Q5 = _F_64(-2.01099218183624371326e-07); /* BE8AFDB7 6E09C32D */
163
164 __float64
_NAME_64(expm1)165 _NAME_64(expm1)(__float64 x)
166 {
167 __float64 y,hi,lo,c,t,e,hxs,hfx,r1;
168 __int32_t k,xsb;
169 __uint32_t hx;
170
171 GET_HIGH_WORD(hx,x);
172 xsb = hx&0x80000000; /* sign bit of x */
173 if(xsb==0) y=x; else y= -x; /* y = |x| */
174 hx &= 0x7fffffff; /* high word of |x| */
175
176 /* filter out huge and non-finite argument */
177 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
178 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
179 if(hx>=0x7ff00000) {
180 __uint32_t low;
181 GET_LOW_WORD(low,x);
182 if(((hx&0xfffff)|low)!=0)
183 return x+x; /* NaN */
184 else return (xsb==0)? x:_F_64(-1.0);/* exp(+-inf)={inf,-1} */
185 }
186 if(x > o_threshold) return __math_oflow (0); /* overflow */
187 }
188 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
189 if(x+tiny<_F_64(0.0)) /* raise inexact */
190 return tiny-one; /* return -1 */
191 }
192 }
193
194 /* argument reduction */
195 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
196 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
197 if(xsb==0)
198 {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
199 else
200 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
201 } else {
202 k = invln2*x+((xsb==0)?_F_64(0.5):_F_64(-0.5));
203 t = k;
204 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
205 lo = t*ln2_lo;
206 }
207 x = hi - lo;
208 c = (hi-x)-lo;
209 }
210 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
211 t = huge+x; /* return x with inexact flags when x!=0 */
212 return x - (t-(huge+x));
213 }
214 else k = 0;
215
216 /* x is now in primary range */
217 hfx = _F_64(0.5)*x;
218 hxs = x*hfx;
219 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
220 t = _F_64(3.0)-r1*hfx;
221 e = hxs*((r1-t)/(_F_64(6.0) - x*t));
222 if(k==0) return x - (x*e-hxs); /* c is 0 */
223 else {
224 e = (x*(e-c)-c);
225 e -= hxs;
226 if(k== -1) return _F_64(0.5)*(x-e)_F_64(-0.5);
227 if(k==1) {
228 if(x < _F_64(-0.25)) return _F_64(-2.0)*(e-(x+_F_64(0.5)));
229 else return one+_F_64(2.0)*(x-e);
230 }
231 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
232 __uint32_t high;
233 y = one-(e-x);
234 GET_HIGH_WORD(high,y);
235 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
236 return y-one;
237 }
238 t = one;
239 if(k<20) {
240 __uint32_t high;
241 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
242 y = t-(e-x);
243 GET_HIGH_WORD(high,y);
244 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
245 } else {
246 __uint32_t high;
247 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
248 y = x-(e+t);
249 y += one;
250 GET_HIGH_WORD(high,y);
251 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
252 }
253 }
254 return y;
255 }
256
257 _MATH_ALIAS_d_d(expm1)
258
259 #endif /* _NEED_FLOAT64 */
260
