2fe8fb192f
There is important information about booting non-ack images in docs/UPDATING. ack/aout-format images can't be built any more, and booting clang/ELF-format ones is a little different. Updating to the new boot monitor is recommended. Changes in this commit: . drop boot monitor -> allowing dropping ack support . facility to copy ELF boot files to /boot so that old boot monitor can still boot fairly easily, see UPDATING . no more ack-format libraries -> single-case libraries . some cleanup of OBJECT_FMT, COMPILER_TYPE, etc cases . drop several ack toolchain commands, but not all support commands (e.g. aal is gone but acksize is not yet). . a few libc files moved to netbsd libc dir . new /bin/date as minix date used code in libc/ . test compile fix . harmonize includes . /usr/lib is no longer special: without ack, /usr/lib plays no kind of special bootstrapping role any more and bootstrapping is done exclusively through packages, so releases depend even less on the state of the machine making them now. . rename nbsd_lib* to lib* . reduce mtree
163 lines
5.1 KiB
C
163 lines
5.1 KiB
C
/* @(#)e_exp.c 5.1 93/09/24 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#include <sys/cdefs.h>
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#if defined(LIBM_SCCS) && !defined(lint)
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__RCSID("$NetBSD: e_exp.c,v 1.11 2002/05/26 22:01:49 wiz Exp $");
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#endif
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/* __ieee754_exp(x)
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* Returns the exponential of x.
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*
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* Method
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* 1. Argument reduction:
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* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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* Given x, find r and integer k such that
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*
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* x = k*ln2 + r, |r| <= 0.5*ln2.
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*
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* Here r will be represented as r = hi-lo for better
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* accuracy.
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*
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* 2. Approximation of exp(r) by a special rational function on
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* the interval [0,0.34658]:
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* Write
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* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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* We use a special Reme algorithm on [0,0.34658] to generate
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* a polynomial of degree 5 to approximate R. The maximum error
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* of this polynomial approximation is bounded by 2**-59. In
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* other words,
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* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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* (where z=r*r, and the values of P1 to P5 are listed below)
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* and
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* | 5 | -59
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* | 2.0+P1*z+...+P5*z - R(z) | <= 2
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* | |
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* The computation of exp(r) thus becomes
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* 2*r
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* exp(r) = 1 + -------
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* R - r
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* r*R1(r)
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* = 1 + r + ----------- (for better accuracy)
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* 2 - R1(r)
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* where
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* 2 4 10
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* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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*
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* 3. Scale back to obtain exp(x):
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* From step 1, we have
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* exp(x) = 2^k * exp(r)
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*
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* Special cases:
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* exp(INF) is INF, exp(NaN) is NaN;
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* exp(-INF) is 0, and
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* for finite argument, only exp(0)=1 is exact.
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*
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* Accuracy:
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* according to an error analysis, the error is always less than
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* 1 ulp (unit in the last place).
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*
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* Misc. info.
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* For IEEE double
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* if x > 7.09782712893383973096e+02 then exp(x) overflow
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* if x < -7.45133219101941108420e+02 then exp(x) underflow
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*
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* Constants:
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* The hexadecimal values are the intended ones for the following
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* constants. The decimal values may be used, provided that the
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* compiler will convert from decimal to binary accurately enough
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* to produce the hexadecimal values shown.
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*/
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#include "math.h"
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#include "math_private.h"
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static const double
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one = 1.0,
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halF[2] = {0.5,-0.5,},
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huge = 1.0e+300,
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twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
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o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
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u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
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ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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-6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
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ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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-1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
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invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
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double
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__ieee754_exp(double x) /* default IEEE double exp */
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{
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double y,hi,lo,c,t;
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int32_t k,xsb;
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u_int32_t hx;
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hi = lo = 0;
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k = 0;
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GET_HIGH_WORD(hx,x);
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xsb = (hx>>31)&1; /* sign bit of x */
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hx &= 0x7fffffff; /* high word of |x| */
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/* filter out non-finite argument */
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if(hx >= 0x40862E42) { /* if |x|>=709.78... */
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if(hx>=0x7ff00000) {
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u_int32_t lx;
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GET_LOW_WORD(lx,x);
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if(((hx&0xfffff)|lx)!=0)
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return x+x; /* NaN */
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else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
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}
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if(x > o_threshold) return huge*huge; /* overflow */
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if(x < u_threshold) return twom1000*twom1000; /* underflow */
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}
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/* argument reduction */
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if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
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hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
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} else {
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k = invln2*x+halF[xsb];
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t = k;
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hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
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lo = t*ln2LO[0];
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}
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x = hi - lo;
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}
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else if(hx < 0x3e300000) { /* when |x|<2**-28 */
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if(huge+x>one) return one+x;/* trigger inexact */
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}
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else k = 0;
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/* x is now in primary range */
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t = x*x;
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c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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if(k==0) return one-((x*c)/(c-2.0)-x);
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else y = one-((lo-(x*c)/(2.0-c))-hi);
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if(k >= -1021) {
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u_int32_t hy;
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GET_HIGH_WORD(hy,y);
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SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
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return y;
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} else {
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u_int32_t hy;
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GET_HIGH_WORD(hy,y);
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SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
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return y*twom1000;
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}
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}
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