1*2fe8fb19SBen Gras /* @(#)s_expm1.c 5.1 93/09/24 */
2*2fe8fb19SBen Gras /*
3*2fe8fb19SBen Gras * ====================================================
4*2fe8fb19SBen Gras * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*2fe8fb19SBen Gras *
6*2fe8fb19SBen Gras * Developed at SunPro, a Sun Microsystems, Inc. business.
7*2fe8fb19SBen Gras * Permission to use, copy, modify, and distribute this
8*2fe8fb19SBen Gras * software is freely granted, provided that this notice
9*2fe8fb19SBen Gras * is preserved.
10*2fe8fb19SBen Gras * ====================================================
11*2fe8fb19SBen Gras */
12*2fe8fb19SBen Gras
13*2fe8fb19SBen Gras #include <sys/cdefs.h>
14*2fe8fb19SBen Gras #if defined(LIBM_SCCS) && !defined(lint)
15*2fe8fb19SBen Gras __RCSID("$NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $");
16*2fe8fb19SBen Gras #endif
17*2fe8fb19SBen Gras
18*2fe8fb19SBen Gras /* expm1(x)
19*2fe8fb19SBen Gras * Returns exp(x)-1, the exponential of x minus 1.
20*2fe8fb19SBen Gras *
21*2fe8fb19SBen Gras * Method
22*2fe8fb19SBen Gras * 1. Argument reduction:
23*2fe8fb19SBen Gras * Given x, find r and integer k such that
24*2fe8fb19SBen Gras *
25*2fe8fb19SBen Gras * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
26*2fe8fb19SBen Gras *
27*2fe8fb19SBen Gras * Here a correction term c will be computed to compensate
28*2fe8fb19SBen Gras * the error in r when rounded to a floating-point number.
29*2fe8fb19SBen Gras *
30*2fe8fb19SBen Gras * 2. Approximating expm1(r) by a special rational function on
31*2fe8fb19SBen Gras * the interval [0,0.34658]:
32*2fe8fb19SBen Gras * Since
33*2fe8fb19SBen Gras * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
34*2fe8fb19SBen Gras * we define R1(r*r) by
35*2fe8fb19SBen Gras * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
36*2fe8fb19SBen Gras * That is,
37*2fe8fb19SBen Gras * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
38*2fe8fb19SBen Gras * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
39*2fe8fb19SBen Gras * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
40*2fe8fb19SBen Gras * We use a special Reme algorithm on [0,0.347] to generate
41*2fe8fb19SBen Gras * a polynomial of degree 5 in r*r to approximate R1. The
42*2fe8fb19SBen Gras * maximum error of this polynomial approximation is bounded
43*2fe8fb19SBen Gras * by 2**-61. In other words,
44*2fe8fb19SBen Gras * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
45*2fe8fb19SBen Gras * where Q1 = -1.6666666666666567384E-2,
46*2fe8fb19SBen Gras * Q2 = 3.9682539681370365873E-4,
47*2fe8fb19SBen Gras * Q3 = -9.9206344733435987357E-6,
48*2fe8fb19SBen Gras * Q4 = 2.5051361420808517002E-7,
49*2fe8fb19SBen Gras * Q5 = -6.2843505682382617102E-9;
50*2fe8fb19SBen Gras * (where z=r*r, and the values of Q1 to Q5 are listed below)
51*2fe8fb19SBen Gras * with error bounded by
52*2fe8fb19SBen Gras * | 5 | -61
53*2fe8fb19SBen Gras * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
54*2fe8fb19SBen Gras * | |
55*2fe8fb19SBen Gras *
56*2fe8fb19SBen Gras * expm1(r) = exp(r)-1 is then computed by the following
57*2fe8fb19SBen Gras * specific way which minimize the accumulation rounding error:
58*2fe8fb19SBen Gras * 2 3
59*2fe8fb19SBen Gras * r r [ 3 - (R1 + R1*r/2) ]
60*2fe8fb19SBen Gras * expm1(r) = r + --- + --- * [--------------------]
61*2fe8fb19SBen Gras * 2 2 [ 6 - r*(3 - R1*r/2) ]
62*2fe8fb19SBen Gras *
63*2fe8fb19SBen Gras * To compensate the error in the argument reduction, we use
64*2fe8fb19SBen Gras * expm1(r+c) = expm1(r) + c + expm1(r)*c
65*2fe8fb19SBen Gras * ~ expm1(r) + c + r*c
66*2fe8fb19SBen Gras * Thus c+r*c will be added in as the correction terms for
67*2fe8fb19SBen Gras * expm1(r+c). Now rearrange the term to avoid optimization
68*2fe8fb19SBen Gras * screw up:
69*2fe8fb19SBen Gras * ( 2 2 )
70*2fe8fb19SBen Gras * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
71*2fe8fb19SBen Gras * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
72*2fe8fb19SBen Gras * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
73*2fe8fb19SBen Gras * ( )
74*2fe8fb19SBen Gras *
75*2fe8fb19SBen Gras * = r - E
76*2fe8fb19SBen Gras * 3. Scale back to obtain expm1(x):
77*2fe8fb19SBen Gras * From step 1, we have
78*2fe8fb19SBen Gras * expm1(x) = either 2^k*[expm1(r)+1] - 1
79*2fe8fb19SBen Gras * = or 2^k*[expm1(r) + (1-2^-k)]
80*2fe8fb19SBen Gras * 4. Implementation notes:
81*2fe8fb19SBen Gras * (A). To save one multiplication, we scale the coefficient Qi
82*2fe8fb19SBen Gras * to Qi*2^i, and replace z by (x^2)/2.
83*2fe8fb19SBen Gras * (B). To achieve maximum accuracy, we compute expm1(x) by
84*2fe8fb19SBen Gras * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
85*2fe8fb19SBen Gras * (ii) if k=0, return r-E
86*2fe8fb19SBen Gras * (iii) if k=-1, return 0.5*(r-E)-0.5
87*2fe8fb19SBen Gras * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
88*2fe8fb19SBen Gras * else return 1.0+2.0*(r-E);
89*2fe8fb19SBen Gras * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
90*2fe8fb19SBen Gras * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
91*2fe8fb19SBen Gras * (vii) return 2^k(1-((E+2^-k)-r))
92*2fe8fb19SBen Gras *
93*2fe8fb19SBen Gras * Special cases:
94*2fe8fb19SBen Gras * expm1(INF) is INF, expm1(NaN) is NaN;
95*2fe8fb19SBen Gras * expm1(-INF) is -1, and
96*2fe8fb19SBen Gras * for finite argument, only expm1(0)=0 is exact.
97*2fe8fb19SBen Gras *
98*2fe8fb19SBen Gras * Accuracy:
99*2fe8fb19SBen Gras * according to an error analysis, the error is always less than
100*2fe8fb19SBen Gras * 1 ulp (unit in the last place).
101*2fe8fb19SBen Gras *
102*2fe8fb19SBen Gras * Misc. info.
103*2fe8fb19SBen Gras * For IEEE double
104*2fe8fb19SBen Gras * if x > 7.09782712893383973096e+02 then expm1(x) overflow
105*2fe8fb19SBen Gras *
106*2fe8fb19SBen Gras * Constants:
107*2fe8fb19SBen Gras * The hexadecimal values are the intended ones for the following
108*2fe8fb19SBen Gras * constants. The decimal values may be used, provided that the
109*2fe8fb19SBen Gras * compiler will convert from decimal to binary accurately enough
110*2fe8fb19SBen Gras * to produce the hexadecimal values shown.
111*2fe8fb19SBen Gras */
112*2fe8fb19SBen Gras
113*2fe8fb19SBen Gras #include "math.h"
114*2fe8fb19SBen Gras #include "math_private.h"
115*2fe8fb19SBen Gras
116*2fe8fb19SBen Gras static const double
117*2fe8fb19SBen Gras one = 1.0,
118*2fe8fb19SBen Gras huge = 1.0e+300,
119*2fe8fb19SBen Gras tiny = 1.0e-300,
120*2fe8fb19SBen Gras o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
121*2fe8fb19SBen Gras ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
122*2fe8fb19SBen Gras ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
123*2fe8fb19SBen Gras invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
124*2fe8fb19SBen Gras /* scaled coefficients related to expm1 */
125*2fe8fb19SBen Gras Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
126*2fe8fb19SBen Gras Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
127*2fe8fb19SBen Gras Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
128*2fe8fb19SBen Gras Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
129*2fe8fb19SBen Gras Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
130*2fe8fb19SBen Gras
131*2fe8fb19SBen Gras double
expm1(double x)132*2fe8fb19SBen Gras expm1(double x)
133*2fe8fb19SBen Gras {
134*2fe8fb19SBen Gras double y,hi,lo,c,t,e,hxs,hfx,r1;
135*2fe8fb19SBen Gras int32_t k,xsb;
136*2fe8fb19SBen Gras u_int32_t hx;
137*2fe8fb19SBen Gras
138*2fe8fb19SBen Gras c = 0;
139*2fe8fb19SBen Gras GET_HIGH_WORD(hx,x);
140*2fe8fb19SBen Gras xsb = hx&0x80000000; /* sign bit of x */
141*2fe8fb19SBen Gras if(xsb==0) y=x; else y= -x; /* y = |x| */
142*2fe8fb19SBen Gras hx &= 0x7fffffff; /* high word of |x| */
143*2fe8fb19SBen Gras
144*2fe8fb19SBen Gras /* filter out huge and non-finite argument */
145*2fe8fb19SBen Gras if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
146*2fe8fb19SBen Gras if(hx >= 0x40862E42) { /* if |x|>=709.78... */
147*2fe8fb19SBen Gras if(hx>=0x7ff00000) {
148*2fe8fb19SBen Gras u_int32_t low;
149*2fe8fb19SBen Gras GET_LOW_WORD(low,x);
150*2fe8fb19SBen Gras if(((hx&0xfffff)|low)!=0)
151*2fe8fb19SBen Gras return x+x; /* NaN */
152*2fe8fb19SBen Gras else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
153*2fe8fb19SBen Gras }
154*2fe8fb19SBen Gras if(x > o_threshold) return huge*huge; /* overflow */
155*2fe8fb19SBen Gras }
156*2fe8fb19SBen Gras if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
157*2fe8fb19SBen Gras if(x+tiny<0.0) /* raise inexact */
158*2fe8fb19SBen Gras return tiny-one; /* return -1 */
159*2fe8fb19SBen Gras }
160*2fe8fb19SBen Gras }
161*2fe8fb19SBen Gras
162*2fe8fb19SBen Gras /* argument reduction */
163*2fe8fb19SBen Gras if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
164*2fe8fb19SBen Gras if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
165*2fe8fb19SBen Gras if(xsb==0)
166*2fe8fb19SBen Gras {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
167*2fe8fb19SBen Gras else
168*2fe8fb19SBen Gras {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
169*2fe8fb19SBen Gras } else {
170*2fe8fb19SBen Gras k = invln2*x+((xsb==0)?0.5:-0.5);
171*2fe8fb19SBen Gras t = k;
172*2fe8fb19SBen Gras hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
173*2fe8fb19SBen Gras lo = t*ln2_lo;
174*2fe8fb19SBen Gras }
175*2fe8fb19SBen Gras x = hi - lo;
176*2fe8fb19SBen Gras c = (hi-x)-lo;
177*2fe8fb19SBen Gras }
178*2fe8fb19SBen Gras else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
179*2fe8fb19SBen Gras t = huge+x; /* return x with inexact flags when x!=0 */
180*2fe8fb19SBen Gras return x - (t-(huge+x));
181*2fe8fb19SBen Gras }
182*2fe8fb19SBen Gras else k = 0;
183*2fe8fb19SBen Gras
184*2fe8fb19SBen Gras /* x is now in primary range */
185*2fe8fb19SBen Gras hfx = 0.5*x;
186*2fe8fb19SBen Gras hxs = x*hfx;
187*2fe8fb19SBen Gras r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
188*2fe8fb19SBen Gras t = 3.0-r1*hfx;
189*2fe8fb19SBen Gras e = hxs*((r1-t)/(6.0 - x*t));
190*2fe8fb19SBen Gras if(k==0) return x - (x*e-hxs); /* c is 0 */
191*2fe8fb19SBen Gras else {
192*2fe8fb19SBen Gras e = (x*(e-c)-c);
193*2fe8fb19SBen Gras e -= hxs;
194*2fe8fb19SBen Gras if(k== -1) return 0.5*(x-e)-0.5;
195*2fe8fb19SBen Gras if(k==1) {
196*2fe8fb19SBen Gras if(x < -0.25) return -2.0*(e-(x+0.5));
197*2fe8fb19SBen Gras else return one+2.0*(x-e);
198*2fe8fb19SBen Gras }
199*2fe8fb19SBen Gras if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
200*2fe8fb19SBen Gras u_int32_t high;
201*2fe8fb19SBen Gras y = one-(e-x);
202*2fe8fb19SBen Gras GET_HIGH_WORD(high,y);
203*2fe8fb19SBen Gras SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
204*2fe8fb19SBen Gras return y-one;
205*2fe8fb19SBen Gras }
206*2fe8fb19SBen Gras t = one;
207*2fe8fb19SBen Gras if(k<20) {
208*2fe8fb19SBen Gras u_int32_t high;
209*2fe8fb19SBen Gras SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
210*2fe8fb19SBen Gras y = t-(e-x);
211*2fe8fb19SBen Gras GET_HIGH_WORD(high,y);
212*2fe8fb19SBen Gras SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
213*2fe8fb19SBen Gras } else {
214*2fe8fb19SBen Gras u_int32_t high;
215*2fe8fb19SBen Gras SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
216*2fe8fb19SBen Gras y = x-(e+t);
217*2fe8fb19SBen Gras y += one;
218*2fe8fb19SBen Gras GET_HIGH_WORD(high,y);
219*2fe8fb19SBen Gras SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
220*2fe8fb19SBen Gras }
221*2fe8fb19SBen Gras }
222*2fe8fb19SBen Gras return y;
223*2fe8fb19SBen Gras }
224