1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/os/ossrv/genericopenlibs/cstdlib/LMATH/S_EXPM1.C Fri Jun 15 03:10:57 2012 +0200
1.3 @@ -0,0 +1,271 @@
1.4 +/* S_EXPM1.C
1.5 + *
1.6 + * Portions Copyright (c) 1993-2005 Nokia Corporation and/or its subsidiary(-ies).
1.7 + * All rights reserved.
1.8 + */
1.9 +
1.10 +
1.11 +/* @(#)s_expm1.c 5.1 93/09/24 */
1.12 +/*
1.13 + * ====================================================
1.14 + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
1.15 + *
1.16 + * Developed at SunPro, a Sun Microsystems, Inc. business.
1.17 + * Permission to use, copy, modify, and distribute this
1.18 + * software is freely granted, provided that this notice
1.19 + * is preserved.
1.20 + * ====================================================
1.21 + */
1.22 +
1.23 +/*
1.24 +FUNCTION
1.25 + <<expm1>>, <<expm1f>>---exponential minus 1
1.26 +INDEX
1.27 + expm1
1.28 +INDEX
1.29 + expm1f
1.30 +
1.31 +ANSI_SYNOPSIS
1.32 + #include <math.h>
1.33 + double expm1(double <[x]>);
1.34 + float expm1f(float <[x]>);
1.35 +
1.36 +TRAD_SYNOPSIS
1.37 + #include <math.h>
1.38 + double expm1(<[x]>);
1.39 + double <[x]>;
1.40 +
1.41 + float expm1f(<[x]>);
1.42 + float <[x]>;
1.43 +
1.44 +DESCRIPTION
1.45 + <<expm1>> and <<expm1f>> calculate the exponential of <[x]>
1.46 + and subtract 1, that is,
1.47 + @ifinfo
1.48 + e raised to the power <[x]> minus 1 (where e
1.49 + @end ifinfo
1.50 + @tex
1.51 + $e^x - 1$ (where $e$
1.52 + @end tex
1.53 + is the base of the natural system of logarithms, approximately
1.54 + 2.71828). The result is accurate even for small values of
1.55 + <[x]>, where using <<exp(<[x]>)-1>> would lose many
1.56 + significant digits.
1.57 +
1.58 +RETURNS
1.59 + e raised to the power <[x]>, minus 1.
1.60 +
1.61 +PORTABILITY
1.62 + Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by
1.63 + the System V Interface Definition (Issue 2).
1.64 +*/
1.65 +
1.66 +/* expm1(x)
1.67 + * Returns exp(x)-1, the exponential of x minus 1.
1.68 + *
1.69 + * Method
1.70 + * 1. Argument reduction:
1.71 + * Given x, find r and integer k such that
1.72 + *
1.73 + * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
1.74 + *
1.75 + * Here a correction term c will be computed to compensate
1.76 + * the error in r when rounded to a floating-point number.
1.77 + *
1.78 + * 2. Approximating expm1(r) by a special rational function on
1.79 + * the interval [0,0.34658]:
1.80 + * Since
1.81 + * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
1.82 + * we define R1(r*r) by
1.83 + * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
1.84 + * That is,
1.85 + * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
1.86 + * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
1.87 + * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
1.88 + * We use a special Reme algorithm on [0,0.347] to generate
1.89 + * a polynomial of degree 5 in r*r to approximate R1. The
1.90 + * maximum error of this polynomial approximation is bounded
1.91 + * by 2**-61. In other words,
1.92 + * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
1.93 + * where Q1 = -1.6666666666666567384E-2,
1.94 + * Q2 = 3.9682539681370365873E-4,
1.95 + * Q3 = -9.9206344733435987357E-6,
1.96 + * Q4 = 2.5051361420808517002E-7,
1.97 + * Q5 = -6.2843505682382617102E-9;
1.98 + * (where z=r*r, and the values of Q1 to Q5 are listed below)
1.99 + * with error bounded by
1.100 + * | 5 | -61
1.101 + * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
1.102 + * | |
1.103 + *
1.104 + * expm1(r) = exp(r)-1 is then computed by the following
1.105 + * specific way which minimize the accumulation rounding error:
1.106 + * 2 3
1.107 + * r r [ 3 - (R1 + R1*r/2) ]
1.108 + * expm1(r) = r + --- + --- * [--------------------]
1.109 + * 2 2 [ 6 - r*(3 - R1*r/2) ]
1.110 + *
1.111 + * To compensate the error in the argument reduction, we use
1.112 + * expm1(r+c) = expm1(r) + c + expm1(r)*c
1.113 + * ~ expm1(r) + c + r*c
1.114 + * Thus c+r*c will be added in as the correction terms for
1.115 + * expm1(r+c). Now rearrange the term to avoid optimization
1.116 + * screw up:
1.117 + * ( 2 2 )
1.118 + * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
1.119 + * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
1.120 + * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
1.121 + * ( )
1.122 + *
1.123 + * = r - E
1.124 + * 3. Scale back to obtain expm1(x):
1.125 + * From step 1, we have
1.126 + * expm1(x) = either 2^k*[expm1(r)+1] - 1
1.127 + * = or 2^k*[expm1(r) + (1-2^-k)]
1.128 + * 4. Implementation notes:
1.129 + * (A). To save one multiplication, we scale the coefficient Qi
1.130 + * to Qi*2^i, and replace z by (x^2)/2.
1.131 + * (B). To achieve maximum accuracy, we compute expm1(x) by
1.132 + * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
1.133 + * (ii) if k=0, return r-E
1.134 + * (iii) if k=-1, return 0.5*(r-E)-0.5
1.135 + * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
1.136 + * else return 1.0+2.0*(r-E);
1.137 + * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
1.138 + * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
1.139 + * (vii) return 2^k(1-((E+2^-k)-r))
1.140 + *
1.141 + * Special cases:
1.142 + * expm1(INF) is INF, expm1(NaN) is NaN;
1.143 + * expm1(-INF) is -1, and
1.144 + * for finite argument, only expm1(0)=0 is exact.
1.145 + *
1.146 + * Accuracy:
1.147 + * according to an error analysis, the error is always less than
1.148 + * 1 ulp (unit in the last place).
1.149 + *
1.150 + * Misc. info.
1.151 + * For IEEE double
1.152 + * if x > 7.09782712893383973096e+02 then expm1(x) overflow
1.153 + *
1.154 + * Constants:
1.155 + * The hexadecimal values are the intended ones for the following
1.156 + * constants. The decimal values may be used, provided that the
1.157 + * compiler will convert from decimal to binary accurately enough
1.158 + * to produce the hexadecimal values shown.
1.159 + */
1.160 +
1.161 +#include "FDLIBM.H"
1.162 +
1.163 +static const double
1.164 +one = 1.0,
1.165 +huge = 1.0e+300,
1.166 +tiny = 1.0e-300,
1.167 +o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
1.168 +ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
1.169 +ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
1.170 +invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
1.171 + /* scaled coefficients related to expm1 */
1.172 +Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
1.173 +Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
1.174 +Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
1.175 +Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
1.176 +Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
1.177 +
1.178 +/**
1.179 +Calculate the exponential of x and subtract 1
1.180 +that is raised to the power x minus 1
1.181 +@return e raised to the power x, minus 1.
1.182 +@param x e's power.
1.183 +*/
1.184 +EXPORT_C double expm1(double x) __SOFTFP
1.185 +{
1.186 + double y,hi,lo,c = 0.0,t,e,hxs,hfx,r1;
1.187 + __int32_t k,xsb;
1.188 + __uint32_t hx;
1.189 +
1.190 + GET_HIGH_WORD(hx,x);
1.191 + xsb = hx&0x80000000; /* sign bit of x */
1.192 + if(xsb==0) y=x; else y= -x; /* y = |x| */
1.193 + hx &= 0x7fffffff; /* high word of |x| */
1.194 +
1.195 + /* filter out huge and non-finite argument */
1.196 + if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
1.197 + if(hx >= 0x40862E42) { /* if |x|>=709.78... */
1.198 + if(hx>=0x7ff00000) {
1.199 + __uint32_t low;
1.200 + GET_LOW_WORD(low,x);
1.201 + if(((hx&0xfffff)|low)!=0)
1.202 + return x+x; /* NaN */
1.203 + else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
1.204 + }
1.205 + if(x > o_threshold) return huge*huge; /* overflow */
1.206 + }
1.207 + if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
1.208 + if(x+tiny<0.0) /* raise inexact */
1.209 + return tiny-one; /* return -1 */
1.210 + }
1.211 + }
1.212 +
1.213 + /* argument reduction */
1.214 + if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
1.215 + if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
1.216 + if(xsb==0)
1.217 + {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
1.218 + else
1.219 + {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
1.220 + } else {
1.221 + k = invln2*x+((xsb==0)?0.5:-0.5);
1.222 + t = k;
1.223 + hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
1.224 + lo = t*ln2_lo;
1.225 + }
1.226 + x = hi - lo;
1.227 + c = (hi-x)-lo;
1.228 + }
1.229 + else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
1.230 + t = huge+x; /* return x with inexact flags when x!=0 */
1.231 + return x - (t-(huge+x));
1.232 + }
1.233 + else k = 0;
1.234 +
1.235 + /* x is now in primary range */
1.236 + hfx = 0.5*x;
1.237 + hxs = x*hfx;
1.238 + r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
1.239 + t = 3.0-r1*hfx;
1.240 + e = hxs*((r1-t)/(6.0 - x*t));
1.241 + if(k==0) return x - (x*e-hxs); /* c is 0 */
1.242 + else {
1.243 + e = (x*(e-c)-c);
1.244 + e -= hxs;
1.245 + if(k== -1) return 0.5*(x-e)-0.5;
1.246 + if(k==1) {
1.247 + if(x < -0.25) return -2.0*(e-(x+0.5));
1.248 + else return one+2.0*(x-e);
1.249 + }
1.250 + if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
1.251 + __uint32_t high;
1.252 + y = one-(e-x);
1.253 + GET_HIGH_WORD(high,y);
1.254 + SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
1.255 + return y-one;
1.256 + }
1.257 + t = one;
1.258 + if(k<20) {
1.259 + __uint32_t high;
1.260 + SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
1.261 + y = t-(e-x);
1.262 + GET_HIGH_WORD(high,y);
1.263 + SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
1.264 + } else {
1.265 + __uint32_t high;
1.266 + SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
1.267 + y = x-(e+t);
1.268 + y += one;
1.269 + GET_HIGH_WORD(high,y);
1.270 + SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
1.271 + }
1.272 + }
1.273 + return y;
1.274 +}