| |
| /* |
| * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. |
| * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. |
| * |
| * This code is free software; you can redistribute it and/or modify it |
| * under the terms of the GNU General Public License version 2 only, as |
| * published by the Free Software Foundation. Oracle designates this |
| * particular file as subject to the "Classpath" exception as provided |
| * by Oracle in the LICENSE file that accompanied this code. |
| * |
| * This code is distributed in the hope that it will be useful, but WITHOUT |
| * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
| * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
| * version 2 for more details (a copy is included in the LICENSE file that |
| * accompanied this code). |
| * |
| * You should have received a copy of the GNU General Public License version |
| * 2 along with this work; if not, write to the Free Software Foundation, |
| * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. |
| * |
| * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA |
| * or visit www.oracle.com if you need additional information or have any |
| * questions. |
| */ |
| |
| /* __ieee754_exp(x) |
| * Returns the exponential of x. |
| * |
| * Method |
| * 1. Argument reduction: |
| * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. |
| * Given x, find r and integer k such that |
| * |
| * x = k*ln2 + r, |r| <= 0.5*ln2. |
| * |
| * Here r will be represented as r = hi-lo for better |
| * accuracy. |
| * |
| * 2. Approximation of exp(r) by a special rational function on |
| * the interval [0,0.34658]: |
| * Write |
| * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... |
| * We use a special Reme algorithm on [0,0.34658] to generate |
| * a polynomial of degree 5 to approximate R. The maximum error |
| * of this polynomial approximation is bounded by 2**-59. In |
| * other words, |
| * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 |
| * (where z=r*r, and the values of P1 to P5 are listed below) |
| * and |
| * | 5 | -59 |
| * | 2.0+P1*z+...+P5*z - R(z) | <= 2 |
| * | | |
| * The computation of exp(r) thus becomes |
| * 2*r |
| * exp(r) = 1 + ------- |
| * R - r |
| * r*R1(r) |
| * = 1 + r + ----------- (for better accuracy) |
| * 2 - R1(r) |
| * where |
| * 2 4 10 |
| * R1(r) = r - (P1*r + P2*r + ... + P5*r ). |
| * |
| * 3. Scale back to obtain exp(x): |
| * From step 1, we have |
| * exp(x) = 2^k * exp(r) |
| * |
| * Special cases: |
| * exp(INF) is INF, exp(NaN) is NaN; |
| * exp(-INF) is 0, and |
| * for finite argument, only exp(0)=1 is exact. |
| * |
| * Accuracy: |
| * according to an error analysis, the error is always less than |
| * 1 ulp (unit in the last place). |
| * |
| * Misc. info. |
| * For IEEE double |
| * if x > 7.09782712893383973096e+02 then exp(x) overflow |
| * if x < -7.45133219101941108420e+02 then exp(x) underflow |
| * |
| * Constants: |
| * The hexadecimal values are the intended ones for the following |
| * constants. The decimal values may be used, provided that the |
| * compiler will convert from decimal to binary accurately enough |
| * to produce the hexadecimal values shown. |
| */ |
| |
| #include "fdlibm.h" |
| |
| #ifdef __STDC__ |
| static const double |
| #else |
| static double |
| #endif |
| one = 1.0, |
| halF[2] = {0.5,-0.5,}, |
| huge = 1.0e+300, |
| twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ |
| o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ |
| u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ |
| ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ |
| -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ |
| ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ |
| -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ |
| invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ |
| P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ |
| P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ |
| P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ |
| P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ |
| P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ |
| |
| |
| #ifdef __STDC__ |
| double __ieee754_exp(double x) /* default IEEE double exp */ |
| #else |
| double __ieee754_exp(x) /* default IEEE double exp */ |
| double x; |
| #endif |
| { |
| double y,hi=0,lo=0,c,t; |
| int k=0,xsb; |
| unsigned hx; |
| |
| hx = __HI(x); /* high word of x */ |
| xsb = (hx>>31)&1; /* sign bit of x */ |
| hx &= 0x7fffffff; /* high word of |x| */ |
| |
| /* filter out non-finite argument */ |
| if(hx >= 0x40862E42) { /* if |x|>=709.78... */ |
| if(hx>=0x7ff00000) { |
| if(((hx&0xfffff)|__LO(x))!=0) |
| return x+x; /* NaN */ |
| else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ |
| } |
| if(x > o_threshold) return huge*huge; /* overflow */ |
| if(x < u_threshold) return twom1000*twom1000; /* underflow */ |
| } |
| |
| /* argument reduction */ |
| if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
| if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ |
| hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; |
| } else { |
| k = invln2*x+halF[xsb]; |
| t = k; |
| hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ |
| lo = t*ln2LO[0]; |
| } |
| x = hi - lo; |
| } |
| else if(hx < 0x3e300000) { /* when |x|<2**-28 */ |
| if(huge+x>one) return one+x;/* trigger inexact */ |
| } |
| else k = 0; |
| |
| /* x is now in primary range */ |
| t = x*x; |
| c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); |
| if(k==0) return one-((x*c)/(c-2.0)-x); |
| else y = one-((lo-(x*c)/(2.0-c))-hi); |
| if(k >= -1021) { |
| __HI(y) += (k<<20); /* add k to y's exponent */ |
| return y; |
| } else { |
| __HI(y) += ((k+1000)<<20);/* add k to y's exponent */ |
| return y*twom1000; |
| } |
| } |
