initial check-in, version 0.5.0 v0.5.0

1 files changed, 155 insertions, 0 deletions

diff --git a/src/math/e_exp.c b/src/math/e_exp.c
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+
+/* @(#)e_exp.c 1.6 04/04/22 */
+/*
+ * ====================================================
+ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ */
+
+/* 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 Remes 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 <math.h>
+#include "math_private.h"
+
+static const double
+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 */
+
+
+double
+exp(double x) /* default IEEE double exp */
+{
+        double y,hi=0.0,lo=0.0,c,t;
+        int32_t k=0,xsb;
+        uint32_t hx;
+
+        GET_HIGH_WORD(hx,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) {
+                uint32_t lx;
+                GET_LOW_WORD(lx,x);
+                if(((hx&0xfffff)|lx)!=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  = (int)(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) {
+            uint32_t hy;
+            GET_HIGH_WORD(hy,y);
+            SET_HIGH_WORD(y,hy+(k<<20));        /* add k to y's exponent */
+            return y;
+        } else {
+            uint32_t hy;
+            GET_HIGH_WORD(hy,y);
+            SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
+            return y*twom1000;
+        }
+}