Atlas - e_exp.c
Home / ext / SDL / src / libm
Lines: 1 | Size: 6136 bytes
[Download] [Show on GitHub] [Search similar files] [Raw] [Raw (proxy)] 
[FILE BEGIN]
1#include "SDL_internal.h"
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
13/* __ieee754_exp(x)
14 * Returns the exponential of x.
15 *
16 * Method
17 *   1. Argument reduction:
18 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
19 *	Given x, find r and integer k such that
20 *
21 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
22 *
23 *      Here r will be represented as r = hi-lo for better
24 *	accuracy.
25 *
26 *   2. Approximation of exp(r) by a special rational function on
27 *	the interval [0,0.34658]:
28 *	Write
29 *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
30 *      We use a special Remez algorithm on [0,0.34658] to generate
31 * 	a polynomial of degree 5 to approximate R. The maximum error
32 *	of this polynomial approximation is bounded by 2**-59. In
33 *	other words,
34 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
35 *  	(where z=r*r, and the values of P1 to P5 are listed below)
36 *	and
37 *	    |                  5          |     -59
38 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
39 *	    |                             |
40 *	The computation of exp(r) thus becomes
41 *                             2*r
42 *		exp(r) = 1 + -------
43 *		              R - r
44 *                                 r*R1(r)
45 *		       = 1 + r + ----------- (for better accuracy)
46 *		                  2 - R1(r)
47 *	where
48 *			         2       4             10
49 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
50 *
51 *   3. Scale back to obtain exp(x):
52 *	From step 1, we have
53 *	   exp(x) = 2^k * exp(r)
54 *
55 * Special cases:
56 *	exp(INF) is INF, exp(NaN) is NaN;
57 *	exp(-INF) is 0, and
58 *	for finite argument, only exp(0)=1 is exact.
59 *
60 * Accuracy:
61 *	according to an error analysis, the error is always less than
62 *	1 ulp (unit in the last place).
63 *
64 * Misc. info.
65 *	For IEEE double
66 *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
67 *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
68 *
69 * Constants:
70 * The hexadecimal values are the intended ones for the following
71 * constants. The decimal values may be used, provided that the
72 * compiler will convert from decimal to binary accurately enough
73 * to produce the hexadecimal values shown.
74 */
75
76#include "math_libm.h"
77#include "math_private.h"
78
79static const double
80one	= 1.0,
81halF[2]	= {0.5,-0.5,},
82huge	= 1.0e+300,
83twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
84o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
85u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
86ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
87	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
88ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
89	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
90invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
91P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
92P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
93P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
94P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
95P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
96
97union {
98	Uint64 u64;
99	double d;
100} inf_union = {
101	SDL_UINT64_C(0x7ff0000000000000)  /* Binary representation of a 64-bit infinite double (sign=0, exponent=2047, mantissa=0) */
102};
103
104double __ieee754_exp(double x)	/* default IEEE double exp */
105{
106	double y;
107	double hi = 0.0;
108	double lo = 0.0;
109	double c;
110	double t;
111	int32_t k=0;
112	int32_t xsb;
113	u_int32_t hx;
114
115	GET_HIGH_WORD(hx,x);
116	xsb = (hx>>31)&1;		/* sign bit of x */
117	hx &= 0x7fffffff;		/* high word of |x| */
118
119    /* filter out non-finite argument */
120	if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
121            if(hx>=0x7ff00000) {
122	        u_int32_t lx;
123		GET_LOW_WORD(lx,x);
124		if(((hx&0xfffff)|lx)!=0)
125		     return x+x; 		/* NaN */
126		else return (xsb==0)? x:0.0;	/* exp(+-inf)={inf,0} */
127	    }
128		#if 1
129		if(x > o_threshold) return inf_union.d; /* overflow */
130		#elif 1
131		if(x > o_threshold) return huge*huge; /* overflow */
132		#else  /* !!! FIXME: check this: "huge * huge" is a compiler warning, maybe they wanted +Inf? */
133		if(x > o_threshold) return INFINITY; /* overflow */
134		#endif
135
136	    if(x < u_threshold) return twom1000*twom1000; /* underflow */
137	}
138
139    /* argument reduction */
140	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
141	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
142		hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
143	    } else {
144		k  = (int32_t) (invln2*x+halF[xsb]);
145		t  = k;
146		hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
147		lo = t*ln2LO[0];
148	    }
149	    x  = hi - lo;
150	}
151	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
152	    if(huge+x>one) return one+x;/* trigger inexact */
153	}
154	else k = 0;
155
156    /* x is now in primary range */
157	t  = x*x;
158	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
159	if(k==0) 	return one-((x*c)/(c-2.0)-x);
160	else 		y = one-((lo-(x*c)/(2.0-c))-hi);
161	if(k >= -1021) {
162	    u_int32_t hy;
163	    GET_HIGH_WORD(hy,y);
164	    SET_HIGH_WORD(y,hy+(k<<20));	/* add k to y's exponent */
165	    return y;
166	} else {
167	    u_int32_t hy;
168	    GET_HIGH_WORD(hy,y);
169	    SET_HIGH_WORD(y,hy+((k+1000)<<20));	/* add k to y's exponent */
170	    return y*twom1000;
171	}
172}
173
174/*
175 * wrapper exp(x)
176 */
177#ifndef _IEEE_LIBM
178double exp(double x)
179{
180	static const double o_threshold =  7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
181	static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
182
183	double z = __ieee754_exp(x);
184	if (_LIB_VERSION == _IEEE_)
185		return z;
186	if (isfinite(x)) {
187		if (x > o_threshold)
188			return __kernel_standard(x, x, 6); /* exp overflow */
189		if (x < u_threshold)
190			return __kernel_standard(x, x, 7); /* exp underflow */
191	}
192	return z;
193}
194#else
195strong_alias(__ieee754_exp, exp)
196#endif
197libm_hidden_def(exp)
198
[FILE END]
(C) 2025 0x4248
(C) 2025 4248 Media and 4248 Systems, All part of 0x4248
See LICENCE files for more information. Not all files are by 0x4248 always check Licencing.