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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13 for performance improvement on pipelined processors.
14 */
15
16 /* expm1(x)
17 * Returns exp(x)-1, the exponential of x minus 1.
18 *
19 * Method
20 * 1. Argument reduction:
21 * Given x, find r and integer k such that
22 *
23 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
24 *
25 * Here a correction term c will be computed to compensate
26 * the error in r when rounded to a floating-point number.
27 *
28 * 2. Approximating expm1(r) by a special rational function on
29 * the interval [0,0.34658]:
30 * Since
31 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
32 * we define R1(r*r) by
33 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
34 * That is,
35 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
36 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
37 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
38 * We use a special Reme algorithm on [0,0.347] to generate
39 * a polynomial of degree 5 in r*r to approximate R1. The
40 * maximum error of this polynomial approximation is bounded
41 * by 2**-61. In other words,
42 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
43 * where Q1 = -1.6666666666666567384E-2,
44 * Q2 = 3.9682539681370365873E-4,
45 * Q3 = -9.9206344733435987357E-6,
46 * Q4 = 2.5051361420808517002E-7,
47 * Q5 = -6.2843505682382617102E-9;
48 * (where z=r*r, and the values of Q1 to Q5 are listed below)
49 * with error bounded by
50 * | 5 | -61
51 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
52 * | |
53 *
54 * expm1(r) = exp(r)-1 is then computed by the following
55 * specific way which minimize the accumulation rounding error:
56 * 2 3
57 * r r [ 3 - (R1 + R1*r/2) ]
58 * expm1(r) = r + --- + --- * [--------------------]
59 * 2 2 [ 6 - r*(3 - R1*r/2) ]
60 *
61 * To compensate the error in the argument reduction, we use
62 * expm1(r+c) = expm1(r) + c + expm1(r)*c
63 * ~ expm1(r) + c + r*c
64 * Thus c+r*c will be added in as the correction terms for
65 * expm1(r+c). Now rearrange the term to avoid optimization
66 * screw up:
67 * ( 2 2 )
68 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
69 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
70 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
71 * ( )
72 *
73 * = r - E
74 * 3. Scale back to obtain expm1(x):
75 * From step 1, we have
76 * expm1(x) = either 2^k*[expm1(r)+1] - 1
77 * = or 2^k*[expm1(r) + (1-2^-k)]
78 * 4. Implementation notes:
79 * (A). To save one multiplication, we scale the coefficient Qi
80 * to Qi*2^i, and replace z by (x^2)/2.
81 * (B). To achieve maximum accuracy, we compute expm1(x) by
82 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
83 * (ii) if k=0, return r-E
84 * (iii) if k=-1, return 0.5*(r-E)-0.5
85 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
86 * else return 1.0+2.0*(r-E);
87 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
88 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
89 * (vii) return 2^k(1-((E+2^-k)-r))
90 *
91 * Special cases:
92 * expm1(INF) is INF, expm1(NaN) is NaN;
93 * expm1(-INF) is -1, and
94 * for finite argument, only expm1(0)=0 is exact.
95 *
96 * Accuracy:
97 * according to an error analysis, the error is always less than
98 * 1 ulp (unit in the last place).
99 *
100 * Misc. info.
101 * For IEEE double
102 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
103 *
104 * Constants:
105 * The hexadecimal values are the intended ones for the following
106 * constants. The decimal values may be used, provided that the
107 * compiler will convert from decimal to binary accurately enough
108 * to produce the hexadecimal values shown.
109 */
110
111 #include <errno.h>
112 #include <float.h>
113 #include <math.h>
114 #include <math-barriers.h>
115 #include <math_private.h>
116 #include <math-underflow.h>
117 #include <libm-alias-double.h>
118 #define one Q[0]
119 static const double
120 huge = 1.0e+300,
121 tiny = 1.0e-300,
122 o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
123 ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
124 ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
125 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
126 /* scaled coefficients related to expm1 */
127 Q[] = { 1.0, -3.33333333333331316428e-02, /* BFA11111 111110F4 */
128 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
129 -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
130 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
131 -2.01099218183624371326e-07 }; /* BE8AFDB7 6E09C32D */
132
133 double
134 __expm1 (double x)
135 {
136 double y, hi, lo, c, t, e, hxs, hfx, r1, h2, h4, R1, R2, R3;
137 int32_t k, xsb;
138 uint32_t hx;
139
140 GET_HIGH_WORD (hx, x);
141 xsb = hx & 0x80000000; /* sign bit of x */
142 if (xsb == 0)
143 y = x;
144 else
145 y = -x; /* y = |x| */
146 hx &= 0x7fffffff; /* high word of |x| */
147
148 /* filter out huge and non-finite argument */
149 if (hx >= 0x4043687A) /* if |x|>=56*ln2 */
150 {
151 if (hx >= 0x40862E42) /* if |x|>=709.78... */
152 {
153 if (hx >= 0x7ff00000)
154 {
155 uint32_t low;
156 GET_LOW_WORD (low, x);
157 if (((hx & 0xfffff) | low) != 0)
158 return x + x; /* NaN */
159 else
160 return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
161 }
162 if (x > o_threshold)
163 {
164 __set_errno (ERANGE);
165 return huge * huge; /* overflow */
166 }
167 }
168 if (xsb != 0) /* x < -56*ln2, return -1.0 with inexact */
169 {
170 math_force_eval (x + tiny); /* raise inexact */
171 return tiny - one; /* return -1 */
172 }
173 }
174
175 /* argument reduction */
176 if (hx > 0x3fd62e42) /* if |x| > 0.5 ln2 */
177 {
178 if (hx < 0x3FF0A2B2) /* and |x| < 1.5 ln2 */
179 {
180 if (xsb == 0)
181 {
182 hi = x - ln2_hi; lo = ln2_lo; k = 1;
183 }
184 else
185 {
186 hi = x + ln2_hi; lo = -ln2_lo; k = -1;
187 }
188 }
189 else
190 {
191 k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
192 t = k;
193 hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
194 lo = t * ln2_lo;
195 }
196 x = hi - lo;
197 c = (hi - x) - lo;
198 }
199 else if (hx < 0x3c900000) /* when |x|<2**-54, return x */
200 {
201 math_check_force_underflow (x);
202 t = huge + x; /* return x with inexact flags when x!=0 */
203 return x - (t - (huge + x));
204 }
205 else
206 k = 0;
207
208 /* x is now in primary range */
209 hfx = 0.5 * x;
210 hxs = x * hfx;
211 R1 = one + hxs * Q[1]; h2 = hxs * hxs;
212 R2 = Q[2] + hxs * Q[3]; h4 = h2 * h2;
213 R3 = Q[4] + hxs * Q[5];
214 r1 = R1 + h2 * R2 + h4 * R3;
215 t = 3.0 - r1 * hfx;
216 e = hxs * ((r1 - t) / (6.0 - x * t));
217 if (k == 0)
218 return x - (x * e - hxs); /* c is 0 */
219 else
220 {
221 e = (x * (e - c) - c);
222 e -= hxs;
223 if (k == -1)
224 return 0.5 * (x - e) - 0.5;
225 if (k == 1)
226 {
227 if (x < -0.25)
228 return -2.0 * (e - (x + 0.5));
229 else
230 return one + 2.0 * (x - e);
231 }
232 if (k <= -2 || k > 56) /* suffice to return exp(x)-1 */
233 {
234 uint32_t high;
235 y = one - (e - x);
236 GET_HIGH_WORD (high, y);
237 SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */
238 return y - one;
239 }
240 t = one;
241 if (k < 20)
242 {
243 uint32_t high;
244 SET_HIGH_WORD (t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
245 y = t - (e - x);
246 GET_HIGH_WORD (high, y);
247 SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */
248 }
249 else
250 {
251 uint32_t high;
252 SET_HIGH_WORD (t, ((0x3ff - k) << 20)); /* 2^-k */
253 y = x - (e + t);
254 y += one;
255 GET_HIGH_WORD (high, y);
256 SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */
257 }
258 }
259 return y;
260 }
261 libm_alias_double (__expm1, expm1)