author Szabolcs Nagy Sat, 17 Nov 2012 22:22:41 +0000 (23:22 +0100) committer Szabolcs Nagy Sat, 17 Nov 2012 22:22:41 +0000 (23:22 +0100)
overflow and underflow was incorrect when the result was not stored.
an optimization for the 0.5*ln2 < |x| < 1.5*ln2 domain was removed.
consistent with the code.

 src/math/exp.c patch | blob | history

index 29bf960..5c0edee 100644 (file)
@@ -25,7 +25,7 @@
*      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
+ *      We use a special Remez 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,
*          | 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)
+ *                              2*r
+ *              exp(r) = 1 + ----------
+ *                            R(r) - r
+ *                                 r*c(r)
*                     = 1 + r + ----------- (for better accuracy)
- *                                2 - R1(r)
+ *                                2 - c(r)
*      where
- *                               2       4             10
- *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
+ *                              2       4             10
+ *              c(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
*
*   3. Scale back to obtain exp(x):
*      From step 1, we have
*
* 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.
+ *          if x >  709.782712893383973096 then exp(x) overflows
+ *          if x < -745.133219101941108420 then exp(x) underflows
*/

#include "libm.h"

static const double
-halF[2] = {0.5,-0.5,},
-huge    = 1.0e+300,
-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 */
+half[2] = {0.5,-0.5},
+ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
+ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
@@ -89,68 +78,56 @@ P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */

-static const volatile double
-twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */
-
double exp(double x)
{
-       double y,hi=0.0,lo=0.0,c,t,twopk;
-       int32_t k=0,xsb;
+       double hi, lo, c, z;
+       int k, sign;
uint32_t hx;

GET_HIGH_WORD(hx, x);
-       xsb = (hx>>31)&1;  /* sign bit of x */
+       sign = hx>>31;
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)  /* NaN */
-                                return x+x;
-                       return xsb==0 ? x : 0.0;  /* exp(+-inf)={inf,0} */
+       /* special cases */
+       if (hx >= 0x40862e42) {  /* if |x| >= 709.78... */
+               if (isnan(x))
+                       return x;
+               if (x > 709.782712893383973096) {
+                       /* overflow if x!=inf */
+                       STRICT_ASSIGN(double, x, 0x1p1023 * x);
+                       return x;
+               }
+               if (x < -745.13321910194110842) {
+                       /* underflow if x!=-inf */
+                       STRICT_ASSIGN(double, x, 0x1p-1000 / -x * 0x1p-1000);
+                       return x;
}
-               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];
-               }
+       if (hx > 0x3fd62e42) {  /* if |x| > 0.5 ln2 */
+               if (hx < 0x3ff0a2b2)  /* if |x| < 1.5 ln2 */
+                       k = 1 - sign - sign; /* optimization */
+               else
+                       k = (int)(invln2*x + half[sign]);
+               hi = x - k*ln2hi;  /* k*ln2hi is exact here */
+               lo = k*ln2lo;
STRICT_ASSIGN(double, x, hi - lo);
-       } else if(hx < 0x3e300000)  {  /* |x| < 2**-28 */
-               /* raise inexact */
-               if (huge+x > 1.0)
-                       return 1.0+x;
-       } else
+       } else if (hx > 0x3e300000)  {  /* if |x| > 2**-28 */
k = 0;
+               hi = x;
+               lo = 0;
+       } else {
+               /* inexact if x!=0 */
+               FORCE_EVAL(0x1p1023 + x);
+               return 1 + x;
+       }

/* x is now in primary range */
-       t  = x*x;
-       if (k >= -1021)
-               INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);
-       else
-               INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0);
-       c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
+       z = x*x;
+       c = x - z*(P1+z*(P2+z*(P3+z*(P4+z*P5))));
+       x = 1 + ((x*c/(2-c) - lo) + hi);
if (k == 0)
-               return 1.0 - ((x*c)/(c-2.0) - x);
-       y = 1.0-((lo-(x*c)/(2.0-c))-hi);
-       if (k < -1021)
-               return y*twopk*twom1000;
-       if (k == 1024)
-               return y*2.0*0x1p1023;
-       return y*twopk;
+               return x;
+       return scalbn(x, k);
}