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1 /* @(#)s_expm1.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
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 $");
18 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
19 // C4756: overflow in constant arithmetic
20 #pragma warning ( disable : 4756 )
24 * Returns exp(x)-1, the exponential of x minus 1.
27 * 1. Argument reduction:
28 * Given x, find r and integer k such that
30 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
32 * Here a correction term c will be computed to compensate
33 * the error in r when rounded to a floating-point number.
35 * 2. Approximating expm1(r) by a special rational function on
36 * the interval [0,0.34658]:
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)
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
58 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
61 * expm1(r) = exp(r)-1 is then computed by the following
62 * specific way which minimize the accumulation rounding error:
64 * r r [ 3 - (R1 + R1*r/2) ]
65 * expm1(r) = r + --- + --- * [--------------------]
66 * 2 2 [ 6 - r*(3 - R1*r/2) ]
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
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 )
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))
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.
104 * according to an error analysis, the error is always less than
105 * 1 ulp (unit in the last place).
109 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
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.
119 #include "math_private.h"
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 */
139 double y
,hi
,lo
,c
,t
,e
,hxs
,hfx
,r1
;
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| */
149 /* filter out huge and non-finite argument */
150 if(hx
>= 0x4043687A) { /* if |x|>=56*ln2 */
151 if(hx
>= 0x40862E42) { /* if |x|>=709.78... */
155 if(((hx
&0xfffff)|low
)!=0)
156 return x
+x
; /* NaN */
157 else return (xsb
==0)? x
:-1.0;/* exp(+-inf)={inf,-1} */
159 if(x
> o_threshold
) return huge
*huge
; /* overflow */
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 */
167 /* argument reduction */
168 if(hx
> 0x3fd62e42) { /* if |x| > 0.5 ln2 */
169 if(hx
< 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
171 {hi
= x
- ln2_hi
; lo
= ln2_lo
; k
= 1;}
173 {hi
= x
+ ln2_hi
; lo
= -ln2_lo
; k
= -1;}
175 k
= (int32_t)(invln2
*x
+((xsb
==0)?0.5:-0.5));
177 hi
= x
- t
*ln2_hi
; /* t*ln2_hi is exact here */
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
));
189 /* x is now in primary range */
192 r1
= one
+hxs
*(Q1
+hxs
*(Q2
+hxs
*(Q3
+hxs
*(Q4
+hxs
*Q5
))));
194 e
= hxs
*((r1
-t
)/(6.0 - x
*t
));
195 if(k
==0) return x
- (x
*e
-hxs
); /* c is 0 */
199 if(k
== -1) return 0.5*(x
-e
)-0.5;
201 if(x
< -0.25) return -2.0*(e
-(x
+0.5));
202 else return one
+2.0*(x
-e
);
204 if (k
<= -2 || k
>56) { /* suffice to return exp(x)-1 */
207 GET_HIGH_WORD(high
,y
);
208 SET_HIGH_WORD(y
,high
+(k
<<20)); /* add k to y's exponent */
214 SET_HIGH_WORD(t
,0x3ff00000 - (0x200000>>k
)); /* t=1-2^-k */
216 GET_HIGH_WORD(high
,y
);
217 SET_HIGH_WORD(y
,high
+(k
<<20)); /* add k to y's exponent */
220 SET_HIGH_WORD(t
,((0x3ff-k
)<<20)); /* 2^-k */
223 GET_HIGH_WORD(high
,y
);
224 SET_HIGH_WORD(y
,high
+(k
<<20)); /* add k to y's exponent */