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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13 for performance improvement on pipelined processors.
17 * Returns exp(x)-1, the exponential of x minus 1.
20 * 1. Argument reduction:
21 * Given x, find r and integer k such that
23 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
25 * Here a correction term c will be computed to compensate
26 * the error in r when rounded to a floating-point number.
28 * 2. Approximating expm1(r) by a special rational function on
29 * the interval [0,0.34658]:
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)
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
51 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
54 * expm1(r) = exp(r)-1 is then computed by the following
55 * specific way which minimize the accumulation rounding error:
57 * r r [ 3 - (R1 + R1*r/2) ]
58 * expm1(r) = r + --- + --- * [--------------------]
59 * 2 2 [ 6 - r*(3 - R1*r/2) ]
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
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 )
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))
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.
97 * according to an error analysis, the error is always less than
98 * 1 ulp (unit in the last place).
102 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
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.
113 #include <math_private.h>
118 o_threshold
= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
119 ln2_hi
= 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
120 ln2_lo
= 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
121 invln2
= 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
122 /* scaled coefficients related to expm1 */
123 Q
[] = { 1.0, -3.33333333333331316428e-02, /* BFA11111 111110F4 */
124 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
125 -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
126 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
127 -2.01099218183624371326e-07 }; /* BE8AFDB7 6E09C32D */
132 double y
, hi
, lo
, c
, t
, e
, hxs
, hfx
, r1
, h2
, h4
, R1
, R2
, R3
;
136 GET_HIGH_WORD (hx
, x
);
137 xsb
= hx
& 0x80000000; /* sign bit of x */
141 y
= -x
; /* y = |x| */
142 hx
&= 0x7fffffff; /* high word of |x| */
144 /* filter out huge and non-finite argument */
145 if (hx
>= 0x4043687A) /* if |x|>=56*ln2 */
147 if (hx
>= 0x40862E42) /* if |x|>=709.78... */
149 if (hx
>= 0x7ff00000)
152 GET_LOW_WORD (low
, x
);
153 if (((hx
& 0xfffff) | low
) != 0)
154 return x
+ x
; /* NaN */
156 return (xsb
== 0) ? x
: -1.0; /* exp(+-inf)={inf,-1} */
160 __set_errno (ERANGE
);
161 return huge
* huge
; /* overflow */
164 if (xsb
!= 0) /* x < -56*ln2, return -1.0 with inexact */
166 math_force_eval (x
+ tiny
); /* raise inexact */
167 return tiny
- one
; /* return -1 */
171 /* argument reduction */
172 if (hx
> 0x3fd62e42) /* if |x| > 0.5 ln2 */
174 if (hx
< 0x3FF0A2B2) /* and |x| < 1.5 ln2 */
178 hi
= x
- ln2_hi
; lo
= ln2_lo
; k
= 1;
182 hi
= x
+ ln2_hi
; lo
= -ln2_lo
; k
= -1;
187 k
= invln2
* x
+ ((xsb
== 0) ? 0.5 : -0.5);
189 hi
= x
- t
* ln2_hi
; /* t*ln2_hi is exact here */
195 else if (hx
< 0x3c900000) /* when |x|<2**-54, return x */
197 t
= huge
+ x
; /* return x with inexact flags when x!=0 */
198 return x
- (t
- (huge
+ x
));
203 /* x is now in primary range */
206 R1
= one
+ hxs
* Q
[1]; h2
= hxs
* hxs
;
207 R2
= Q
[2] + hxs
* Q
[3]; h4
= h2
* h2
;
208 R3
= Q
[4] + hxs
* Q
[5];
209 r1
= R1
+ h2
* R2
+ h4
* R3
;
211 e
= hxs
* ((r1
- t
) / (6.0 - x
* t
));
213 return x
- (x
* e
- hxs
); /* c is 0 */
216 e
= (x
* (e
- c
) - c
);
219 return 0.5 * (x
- e
) - 0.5;
223 return -2.0 * (e
- (x
+ 0.5));
225 return one
+ 2.0 * (x
- e
);
227 if (k
<= -2 || k
> 56) /* suffice to return exp(x)-1 */
231 GET_HIGH_WORD (high
, y
);
232 SET_HIGH_WORD (y
, high
+ (k
<< 20)); /* add k to y's exponent */
239 SET_HIGH_WORD (t
, 0x3ff00000 - (0x200000 >> k
)); /* t=1-2^-k */
241 GET_HIGH_WORD (high
, y
);
242 SET_HIGH_WORD (y
, high
+ (k
<< 20)); /* add k to y's exponent */
247 SET_HIGH_WORD (t
, ((0x3ff - k
) << 20)); /* 2^-k */
250 GET_HIGH_WORD (high
, y
);
251 SET_HIGH_WORD (y
, high
+ (k
<< 20)); /* add k to y's exponent */
256 weak_alias (__expm1
, expm1
)
257 #ifdef NO_LONG_DOUBLE
258 strong_alias (__expm1
, __expm1l
)
259 weak_alias (__expm1
, expm1l
)